Conversion of Cold Dark Matter Axions to Photons in Astrophysical Magnetic Fields

CONVERSION OF COLD DARK MATTER AXIONS TO PHOTONS IN ASTROPHYSICAL MAGNETIC FIELDS

A thesis submitted to the University of Manchester for the degree of Master of Science in the Faculty of Engineering and Physical Sciences

April 2019

By Sankarshana Srinivasan School of Physics and Astronomy Contents

Abstract8

Declaration9

Acknowledgements 12

1 Introduction 14 1.1 History...... 15 1.2 The Current Picture of Cosmology...... 18 1.2.1 Evolution of the Universe...... 18 1.3 Problems with ΛCDM...... 25 1.3.1 Horizon and Flatness Problems - Inﬂation...... 25 1.4 Evidence for Dark Matter...... 30 1.4.1 Galactic Rotation Curves...... 30 1.4.2 Galaxy Clusters...... 32 1.4.3 Cosmological Evidence...... 34 1.5 Dark Matter Candidates...... 37 1.5.1 Weakly Interacting Massive Particles (WIMPS)...... 37

2 1.5.2 Primordial Black Holes and Massive Astrophysical Com- pact Halo Objects (MACHOs)...... 38 1.5.3 Axions...... 40

2 Axion Models and Constraints 43 2.1 What are axions?...... 44 2.2 Properties and models of axions...... 46 2.3 Experimental Constraints...... 49 2.3.1 Axion Helioscopes...... 49 2.3.2 Haloscope Experiments...... 52 2.3.3 Laboratory based experiments...... 57 2.3.4 Critical Analysis...... 61 2.4 Theoretical Constraints...... 63 2.4.1 Astrophysical constraints...... 63 2.4.2 Constraints from Cosmology...... 66 2.5 Discussion...... 72 2.5.1 This Project...... 73

3 Decay in Cosmological Sources 76 3.1 Introduction...... 77 3.2 Evaluation of Flux and Intensity...... 78 3.2.1 Physical Observables and Quantities...... 78 3.2.2 Intensity and Flux...... 81 3.3 Astrophysical Magnetic Fields...... 87 3.4 Detectability...... 89 3.4.1 Galactic Centre...... 91 3.4.2 Virialised Objects...... 93 3.5 Discussion...... 100

3 4 The Axion-Photon Decay in Neutron Stars 107 4.1 Introduction...... 108 4.2 Modiﬁcation of Maxwell’s Equations...... 109 4.3 Goldreich-Julian Calculation...... 111 4.4 General Formalism...... 117 4.5 Critical Analysis of Literature...... 121 4.5.1 Raﬀelt and Stodolsky...... 121 4.5.2 Hook et al...... 125 4.5.3 The Landau-Zener Solution...... 130 4.6 Discussion...... 137

5 Conclusions 143

Bibliography 149

4 List of Tables

2.1 Critical comparison of axion experiments...... 62

3.1 Table of distances, velocity widths and masses of local dwarf galaxies 96 3.2 Table of distances, velocity widths and masses of clusters of galaxies.101 3.3 Table of integration times from beam mass estimation...... 105

5 List of Figures

1.1 The scale factor as a function of time...... 20 1.2 The energy budget of the Universe...... 23 1.3 History of the Universe (WMAP)...... 29 1.4 The rotation curve of the Andromeda (M31) Galaxy...... 31 1.5 The Bullet Cluster...... 34 1.6 The temperature angular power spectrum from Planck...... 36 1.7 Constraints on the WIMP parameter space by PandaX-II.... 38 1.8 Limits on primordial black holes...... 40

2.1 Axion parameter space...... 45 2.2 Axion-Photon decay mechanisms...... 48 2.3 The CAST helioscope at CERN...... 51 2.4 CAST exclusion plot...... 52 2.5 The ADMX experiment...... 53 2.6 ADMX limit on CDM axions...... 54 2.7 The CROWS Experiment...... 57 2.8 Representation of photon regeneration experiments...... 59 2.9 Limits on axion parameter space by photon regeneration experiments...... 60 2.10 Axion cooling constraints...... 66

6 2.11 Axion couplings to gluons and pions...... 70 2.12 Summary of axion constraints...... 73

3.1 Magnetic ﬁeld power spectrum of the Coma cluster...... 88 3.2 Radio signal detection schematic...... 90 3.3 Detection regimes for virialised objects...... 94 3.4 Integration time trends for resolved and unresolved detections.. 103 3.5 1-σ sensitivities from Virgo observations...... 106

4.1 Probability of axion-photon conversion...... 130 4.2 1 − σ sensitivities from neutron star observations...... 140

7 Abstract

The cold dark matter (CDM) problem is one of the important unsolved problems in cosmology today. The axion is one of the well motivated CDM candidates whose weak spontaneous decay to photons is enhanced in the presence of magnetic fields. This thesis examines the prospects of detecting the axion-photon decay in astrophysical magnetic fields using radio telescopes, with special reference to cosmological objects and neutron stars. Cosmological objects are found to be ill- suited to the detection of the enhanced decay because of the large scale structure of magnetic fields in such objects. However, nearby massive galaxy clusters might offer a chance of detection of the spontaneous decay, although this requires a clear understanding of the density profile of such clusters. Neutron stars and their magnetospheres are home to a resonance effect that could be one of the best chances of detection in future searches. We develop a general formalism that allows us to determine the different assumptions that lead to different results in the literature. We also examine the detectability of the flux from this resonant decay and find that radio telescopes are sensitive enough to probe the regions of CDM axion parameter space hitherto unexplored.

8 Declaration

No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualiﬁcation of this or any other university or other institution of learning.

9 Copyright

i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the Copyright) and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trade marks and other intellectual property (the Intellectual Property) and any reproductions of copyright works in the thesis, for example graphs and tables (Reproductions), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property

10 and/or Reproductions described in it may take place is available in the University IP Policy (see http://www.campus.manchester.ac.uk/medialibrary/policies/intellectual- property.pdf), in any relevant Thesis restriction declarations deposited in the Uni- versity Library, The University Librarys regulations (see http://www.manchester.ac.uk/library/ aboutus/regulations) and in The University’s policy on presentation of The- ses.

11 Acknowledgements

I would like to thank my supervisor Professor Richard Battye for the support and knowledge he has provided me over the course of my MSc. I would also like to thank Dr Francesco Pace for his invaluable help, patience, guidance and support. Special thanks to Susmita Sett, Dirk Scholte, Anthony Gilfelon, David Whitworth, Tracy Garratt, Suheil Harjani and Thomas Peasley for their help and encouragement. Finally, I thank my parents for their constant support and faith.

Chapter 1

Introduction to Cosmology and Dark Matter

14 1.1. HISTORY 15

1.1 History

Cosmology, the study of the origin, evolution and fate of the universe can be traced back to ancient times. The word cosmology comes from the Greek words Kosmos, which means “world” and -logia which means “study of”. The first cosmological theories can be traced back to Mesopotamia and India. Mesopotamian cosmology describes a flat circular Earth surrounded by a cosmic ocean. The Rigveda has some verses which detail a qualitative cosmological theory that describes a cyclic universe expanding from a point called the “bindu” and collapsing back to it. In the West, the earliest cosmological theories were geocentric theories proposed by Greek philosophers such as Ptolemy and Aristotle. They observed the sun rising in the East and setting in the West, as well as observed the movement of constellations in the night sky through the year. Thus, this system, dubbed the “Ptolemaic System” was the generally accepted one as it fit “observations” until the 16th century. Interestingly, the Greek astronomer Aristarchus had already presented a heliocentric model with static stars and the earth orbiting the sun. Unfortunately, his ideas were rejected for Aristotle’s.

With the dawn of the 16th century came the first heliocentric cosmological model. This change was heralded by Nicholas Copernicus and his ’Copernican Principle’, which stated that the movement of the celestial bodies mimic the movement of objects on Earth. By this time, many Islamic astronomers and scholars had already rejected the stationary theories of Aristotle and Ptolemy. However, the dawn of physical cosmology came with Johannes Kepler and Isaac Newton, who both described mathematical laws based on empirical physical observations. Newton’s laws of gravitation and motion were the first mathematical 16 CHAPTER 1. INTRODUCTION treatments of Cosmology that allowed one to check if ideas fit observations uniformly [1].

Modern cosmology began with Einstein’s Special Relativity in 1905 [2], which first established the existence of spacetime and led Einstein to seek a modification of Newton’s Law of Gravity. Interestingly, three years after Einstein’s paper was published, Henrietta Leavitt published the Period-Luminosity relationship for Cepheid-Variable stars [3], later used by Edwin Hubble to discover the expansion of the Universe [4]. Ten years after Special Relativity, Einstein published his General Theory of Relativity (GR), which succeeded Newton’s law as the universally accepted theory of Gravity [5]. The solution of Einstein’s equations assuming an isotropic and homogeneous universe is encoded in the Friedmann- Lemaitre-Robertson-Walker (FLRW) metric [6–9]. The FLRW metric along with the field equations of GR form the mathematical background for all of modern cosmology.

Meanwhile, the physicists Hans Bethe, Ralph Alpher and George Gamow examined element synthesis in an expanding, cooling universe and proposed that rapid neutron capture can be a mechanism by which elements may be created in the Early Universe [10]. George Gamow also predicted the presence of the Cosmic Microwave Background (CMB) from the behaviour of primordial radiation [11], later serendipitously discovered by Penzias and Wilson [12]. This was viewed by Robert Dicke, David Todd Wilkinson, James Peebles and Peter Roll as a relic of the big bang [13]. Fred Hoyle along with William Fowler and Robert Wagoner later proved that the big bang model produced the right abundances of Helium and Deuterium [14]. 1.1. HISTORY 17

The model of an expanding universe assumes the universe began in a hot, dense state with all its constituents in a plasma (thermal bath), dubbed “the big bang” [15]. As the universe cooled and expanded, the different particles constitut- ing the universe escaped this plasma in a process called “freeze-out”. The CMB is an example of relativistic freeze-out of photons. This picture has been confirmed from a variety of observations and theoretical explanations. The current cosmological model is called the ΛCDM model. The Λ is the cosmological constant, introduced into the field equations of GR by Einstein, in a bid to maintain a static Universe. Such a term produces an accelerated expansion for the Universe, which was observationally confirmed by two groups headed by Brian Schmidt, Adam Riess and Saul Perlmutter [16, 17]. This accelerated expansion is interpreted as being due to a mysterious kind of energy-density called dark energy, described as a fluid with an unusual equation of state, i.e., with negative pressure. CDM represents cold dark matter, a hitherto unknown form of non-relativistic (hence cold) matter whose existence is inferred purely from its gravitational interactions. These are the two “dark” components of the universe. Cold dark matter was first hypothesised by the Swiss astronomer Fritz Zwicky [18]. The fundamental idea was to examine the speed of objects orbiting galaxies and infer a mass of the galactic matter from the virial theorem. This mass estimate always exceeded the luminous mass estimates. Thus, the matter in the universe is dominated by an unknown form of “dark” matter. This picture was confirmed observationally by Vera Rubin who was the first astronomer to establish that galaxy rotation curves are flatter than they should be [19]. 18 CHAPTER 1. INTRODUCTION

1.2 The Current Picture of Cosmology

1.2.1 Evolution of the Universe

The motivation for modifying Newton’s law of Gravitation comes from Special Relativity. Newton’s law describes an action-at-a-distance force. Therefore, physical information travels instantaneously between objects. This is in contradiction with Special Relativity. In Special Relativity, no physical information can propagate faster than the speed of light in vacuum. This is in concordance with the principle of relativity, that states that the laws of physics are the same in all inertial frames of reference. Thus, Einstein was seeking to formulate a theory of gravitation that accounts for special relativity. The fundamental idea that forms the basis for General Relativity is the Strong Equivalence Principle (SEP). The SEP states that a uniformly accelerating reference frame is physically indistinguishable from a gravitational field. Therefore, if one hypothesises that all gravitational fields are simply observers uniformly accelerating in spacetime, it automatically follows that mass causes curvature of spacetime. The field equations of General Relativity can be written compactly as

1 8πG G = R − g R = − T , (1.1) µν µν 2 µν c4 µν

where Gµν is the Einstein tensor, Rµν is the Ricci tensor, which is the contraction

ρ of the Riemann Tensor Rµν = Rµρν, which describes the curvature of spacetime,

µν R = g Rµν is the Ricci scalar, is gµν is the metric, which describes distances in spacetime, Tµν is the energy-momentum tensor, G is the universal gravitational constant and c is the speed of light. These ﬁeld equations can simply 1.2. THE CURRENT PICTURE OF COSMOLOGY 19 be interpreted as energy density leading to curvature of spacetime. Since Spe- cial Relativity established that energy and mass are equivalent, the curvature of spacetime can also be caused by diﬀerent forms of energy. The spherical sym- metric vacuum solution to Einstein’s equations was developed by Schwarzschild in 1916 [20].

If one were to impose that the Universe is homogeneous and isotropic, one then obtains the FLRW metric

dr2 ds2 = c2dt2 − a(t)2 + r2dθ2 + r2 sin2 θdφ2 , (1.2) 1 − kr2 where ds2 is the distance measure in the metric and a(t) is called the scale factor. It is introduced to model the expansion of the universe, which results in distances between two spacetime points increasing with time. The usual normalisation for a(t) is a(t0) = 1, where t0 is the current time. One can deﬁne a(t) to have units of length, which results in r and k being dimensionless. This results in the following normalisation for k(r), a constant that tells us the geometry of the universe, and consequently the fate of the universe (see Fig. 1.1).

  1 if universe has spherical geometry and finite ,   k(r) = 0 if universe has flat geometry and is infinite , (1.3)    −1 if universe has hyperbolic geometry and is infinite .

Substituting the FLRW metric into Einstein’s equations and solving them, we obtain the two evolution equations for the universe, namely, the Friedmann equation and the Raychaudhuri equation. The Friedmann equation is obtained from the 20 CHAPTER 1. INTRODUCTION

k=0 k=1 k=-1 a(t)

t Figure 1.1: The scale factor as function of time. The k = 1 corresponds to a closed Universe with spherical geometry. The k = −1 corresponds to an open Universe with hyperbolic geometry. The k = 0 case corresponds to the ﬂat, open Universe. This case is corroborated by CMB measurements [21,22].

00-component of the Ricci tensor

a˙ 2 8πGρ kc2 H2 = = + , (1.4) a 3 a2

a˙ where H = a is the Hubble constant and ρ is the density associated to the energy- momentum ﬂuid of the universe, and the dot represents the derivative with respect to time. The Raychaudhuri equation is obtained from the Friedmann equation and the trace of Einstein’s ﬁeld equations

a¨ 4πG 3p = − ρ + , (1.5) a 3 c2 where p is the pressure of the energy-momentum ﬂuid. Finally, we also have the 1.2. THE CURRENT PICTURE OF COSMOLOGY 21

µν energy conservation equation, derived from the statement ∇µT = 0 (energy- momentum is locally conserved in the evolution of the universe)

p ρ˙ = −3H(ρ + ) . (1.6) c2

Assuming that the energy-momentum of the universe is a perfect fluid with a particular equation of state of the form P = f(ρ), we now have a closed system of equations, which we can solve to find the evolution of the Universe. It is very useful to introduce the critical density parameter, defined as the density that just closes the universe today 3H2 ρ = 0 . (1.7) c,0 8πG

Using the critical density parameter, we can express the contribution of the different energy-momentum constituents, i.e., matter, radiation, dark energy and curvature Ω Ω Ω H2 = H2 m,0 + r,0 + κ,0 + Ω , (1.8) 0 a3 a4 a2 Λ,0

ρi,0 where Ω , = is the density parameter for fluid i today. The matter density i 0 ρc,0 1 ρm scales as a3 since matter has zero pressure, while the radiation density ρr 1 1 2 scales as a4 which reflects the radiation equation of state P = 3 ρc . This means that relativistic energy density diminishes faster than the non-relativistic energy density, because of relativistic momentum, which is associated to wavelength by the De-Broglie hypothesis. The cosmological constant density doesn’t scale with the scale factor because of the negative pressure p = −ρc2 , that arises from the introduction of the cosmological constant Λ into Einstein’s field equations. The curvature density doesn’t reflect an actual density-pressure equation of state but is a convenient way to model the effect of curvature since we can then express all 22 CHAPTER 1. INTRODUCTION

the density parameters as a sum

Ωm + Ωr + ΩΛ + Ωκ = 1 . (1.9)

The current picture of the Universe, see Fig.(1.2), comes from the results of the Planck mission that inferred cosmological parameters from CMB measurements [21]. A succinctly summarised history of the universe is as follows. The Universe began with an inﬂationary phase of accelerated expansion, in which the primordial perturbations that formed the seeds for structure formation and the temperature ﬂuctuations observed in the CMB originated. This was followed by a reheating mechanism through which the Universe transitioned into radiation

1 dominated expansion during which the a4 expansion of the Universe proceeded, and the universe began to cool. Particles began to decouple from the hot plasma, the CMB being an example of relativistic decoupling (of photons). At appropriate thermal conditions, nucleosynthesis occurred, producing Hydrogen, Deu- terium and Helium. As the Universe continued to expand and cool, the matter

1 component began to dominate, corresponding to the a3 term. During the matter dominated epoch, signiﬁcant structure formation occurred [23]. Currently, the universe is in a state of accelerated expansion, where ΩΛ dominates. This information can be gleaned from the CMB observations made by Planck. According to Planck, Ωm,0 ≈ 0.315 , ΩΛ,0 ≈ 0.684 , Ωκ,0 + Ωr,0 ≤ 0.001 [21, 22]. Therefore, observations indicate to a large precision that our Universe has a ﬂat geometry,

which implies that Ωκ = 0 throughout the history of the Universe (see section 1.3.1, the ﬂatness problem). 1.2. THE CURRENT PICTURE OF COSMOLOGY 23

Dark Energy

70%

4% Baryons + Radiation 26%

Cold Dark Matter

Figure 1.2: The energy budget of the Universe.

Dark Energy and the Cosmological Constant Now, the two main components of the ΛCDM model are described in brief. It is these two components that dominate the energy budget of the Universe [21], with ΩΛ,0 + ΩCDM,0 ≈ 0.95. Firstly, we discuss the cosmological constant and dark energy.

Einstein initially introduced the idea of the cosmological constant as a means to make the universe static, i.e.,

Gµν + Λgµν = −κTµν , (1.10)

8πG where we have set κ = c4 . By introducing this cosmological constant, Einstein sought to “balance” the effect of gravity, resulting in a static Universe. However, other than this (physically unmotivated) reason, there was no physical reason to include it in the theory. Einstein later regarded this as his biggest blunder and retracted the idea. However, the discovery of the accelerated expansion of the Universe has revealed that Einstein’s field equations and a cosmological constant are required to accurately describe the Universe as we observe it. The 24 CHAPTER 1. INTRODUCTION cosmological constant mimics a fluid that obeys an equation of state P = wρc2, where w = −1. Extensive research is done on the properties of the cosmological constant, especially with reference to whether its value changes over time and the cosmological constant problem [24]. Quantum field theory describes spacetime with quantum fluctuations that would add to the cosmological constant. However this theoretical value is many orders of magnitude off the measured value. This is called the cosmological constant problem [23,24].

Dark Matter In addition to evidence from galaxy rotation curves and gravitational lensing [25], cold dark matter also plays a very important role in structure formation. We employ perturbation theory to study the Early Universe and the evolution of structures, with the assumption that at the big bang epoch, small perturbations in matter and radiation were generated. These perturbations grew as the universe expanded. Cold dark matter is the primary source that drives structure formation, as it is the matter perturbation that grows with the scale factor in the evolution of the universe [23]. 1.3. PROBLEMS WITH ΛCDM 25

1.3 Problems with ΛCDM

ΛCDM has been extremely successful in describing the universe as a whole. How- ever, there are some problems with the theory that are still unsolved. These are

•The Cosmological Constant Problem

•The Horizon Problem

•The Flatness Problem

•The nature of dark matter and dark energy.

We have already discussed the cosmological constant problem. Other than the observational evidence that we have from the accelerated expansion of the Uni- verse, we do not have any fundamental motivation in introducing the cosmological constant and therefore do not understand the underlying mechanism that results in the negative pressure of the dark energy ﬂuid. Similarly, although we can infer the existence of dark matter and measure its inﬂuence, we do not understand how this form of matter ties in to our understanding of fundamental physics and the Standard Model of particle physics (SM). Solving the mystery of the nature of dark matter and dark energy constitutes the future of cosmology and fundamental physics. Apart from this, we also have problems concerning the initial conditions of the ΛCDM model. These problems are now discussed.

1.3.1 Horizon and Flatness Problems - Inﬂation

We now discuss the Horizon and Flatness problems. The cosmological horizon is the distance travelled by a light signal emitted at the big bang epoch (t = 0). Two 26 CHAPTER 1. INTRODUCTION

regions between which physical information can be exchanged in a time period smaller than or equal to the age of the universe are called causally connected regions. Thus, the horizon size describes the size of a causally connected region, assuming the big bang happened at zero time. The ratio of the size of the horizon

now and at time trec (time of recombination) is given by [26]

2 l0 = 1 + z(trec) ≈ 1000 . (1.11) ltrec

Therefore, there are around a thousand regions in the universe that have never been in causal contact. Despite this all of the universe has about the same temperature to within one part in 104, as evidenced by the CMB. This is called the horizon problem.

From Planck, we know that |Ωκ < 0.02| [22]. We also know that its time evolution is given by 1 Ω (t) ∝ . (1.12) κ a2(t)H2(t)

Therefore, Ω (t ) κ 0 ' 1060 . (1.13) Ωκ(trec)

60 This means that the value of Ωκ would have to be of the order of 10− in the beginning, a very unnatural ﬁne tuning, indeed! This is called the ﬂatness problem.

The reason for the breakdown of our understanding at the earliest epochs could be due to our lack of knowledge of the behaviour of Quantum Gravity, whose effects must have been signiﬁcant at the earliest stages. However, we do have a theoretical framework that provides a solution to these problems, called inﬂation. 1.3. PROBLEMS WITH ΛCDM 27

The idea of inflation is that the universe underwent a period of accelerated expansion before the hot expansion phase [see Fig.(1.3)], similar to the situation right now, powered by a scalar field that dominates the energy-momentum of the universe. Therefore, the radiation-domination period would be preceded by this inflationary period. To demonstrate this, consider a scalar field minimally coupled to gravity

Z Z 4 √ 4 √ µν S = d x −gL = d x −g (g ∂µφ∂νφ − V (φ)) , (1.14) where φ is the scalar ﬁeld. This leads to the following equation of motion

φ¨ + 3Hφ˙ + V 0(φ) = 0 , (1.15) where the prime denotes the derivative with respect to φ. The energy momentum tensor is given by 2 δS T = √ = ∂ φ∂ φ − g L . (1.16) µν −g δgµν µ ν µν

Interpreting Tµν as an ideal ﬂuid, we have

1 ρ = φ˙2 + V (φ) , (1.17) 2 p 1 = φ˙2 − V (φ) . (1.18) c2 2

Assuming that the Hubble friction term dominates

φ¨ 1 , (1.19) 3Hφ˙ 28 CHAPTER 1. INTRODUCTION

and that the kinetic energy is negligible when compared to the potential energy φ˙2 1 , (1.20) 2V

we obtain that P ≈ −ρc2 . (1.21)

This is identical to the cosmological constant equation of state that causes an accelerated expansion of the universe. The conditions (1.19- 1.20) are collectively called the slow roll approximation. This is quantiﬁed by the slow-roll parameters

M 2 V 2 = Pl 0 , (1.22) 16π V M 2 V η = pl 00 , (1.23) 8π V

2 ~c where MPl = 8πG is the reduced Planck mass. If 1 and η 1, we are in the slow-roll regime. This is an example of an inflationary model, i.e., slow- roll inflation. If the inflationary period is long enough, i.e., if there is enough expansion, and that there exists a process by which the universe is reheated after the inflationary expansion just before radiation domination (this is a non trivial assumption), the horizon and flatness problems are solved since inflation takes the origin of conformal time far back in the past, increasing the size of a causally connected region. Therefore, we have a physical mechanism that explains the flatness problem. Unfortunately, in exchange for that, we now have to deal with another fine tuning, the potential φ must be exceedingly flat to be consistent with CMB observations. This results in another fine tuning problem for the coupling between the potential and φ [27]. 1.3. PROBLEMS WITH ΛCDM 29

Figure 1.3: A concise history of the Universe as presented by the Wilkinson Microwave Anisotropy Probe (WMAP) probe team.

Therefore, it is clear that much work is needed to reconcile these problems with our current cosmological model. This work aims to explore one of these problems, namely the dark matter problem, by the study of a dark matter candidate particle called the axion and the possible ways by which it could be detected in cosmological/astrophysical standpoint. Before the axion is introduced and the relevant models are discussed, a brief discussion on the evidence for dark matter is presented. 30 CHAPTER 1. INTRODUCTION

1.4 Evidence for Dark Matter

The dark matter hypothesis is backed up by evidence from multiple disciplines over a wide range of scales, from galactic studies, gravitational lensing and CMB measurements. From galactic data to cosmological CMB data, we see that a wide range of phenomena would be unexplained if not for dark matter.

1.4.1 Galactic Rotation Curves

The luminous matter in galaxies is concentrated in the centre and starts to decrease in density as one moves radially away from the centre. Therefore, the velocity of stars at the “edge” (far away from the dense luminous centre) of the galaxy must be lower than the stars closer in. However, when the velocity is plotted against distance to the centre in a so-called rotation curve, the tail of the plot remains flat, or sometimes even increases [see Fig.(1.4)]. This fact was first established by the astronomers Vera Rubin and Kent Ford [19]. Rubin was controversially not awarded the Nobel prize for her discovery of perhaps the con- clusive proof of the existence of dark matter. Without hypothesising the existence of a halo of dark matter surrounding the luminous matter in the galaxy, it is very difficult to explain the asymptotic behaviour of these rotation curves. 1.4. EVIDENCE FOR DARK MATTER 31

Figure 1.4: The Rotation Curve of the Andromeda (M31) Galaxy. The purple points are emission line data in the outer parts from Babcock 1939 [28]. The black points are from Rubin and Ford’s original paper in 1970 [19]. The red points are the 21-cm HI line data from Roberts and Whitehurst 1975 [29]. The green points are 21-cm HI line data from Carignan et al [30]. The black solid line corresponds to the rotation curve of an exponential disc with a scale length according to the value given in Freeman 1970 [31], suitably scaled in velocity. 21-cm data demonstrate clearly the mass discrepancy in the outer parts. Figure courtesy of Albert Bosma [32].

By 1978, work done by Albert Bosma for his PhD thesis [32], as well as Rubin, Thonnard and Kent had established that velocities remained ﬂat out to large radii, and that the missing mass problem was consistently present everywhere [33]. Using measurements of the Milky Way’s rotation curve, the local dark matter density has been measured in a number of studies, although these results are strongly dependent on assumptions one makes on the actual shape of the galaxy’s dark matter halo [34–37]. 32 CHAPTER 1. INTRODUCTION

1.4.2 Galaxy Clusters

As mentioned before, the Swiss Astronomer Fritz Zwicky was the ﬁrst to hy- pothesise “missing matter” to explain astronomical observations. Zwicky studied the redshifts of various galaxy clusters as published by Hubble and Humason in 1931 [38,39]. Zwicky noticed that there was a rather large velocity distribution in

1 these measurements, exceeding 2000 kms− . He then applied the virial theorem to estimate the mass of these clusters and found an abnormally high mass to light ratio [40] of about 500. In this estimate, Zwicky relied on a Hubble constant value

1 1 inferred from Hubble and Humason’s paper, which was 558 km s− Mpc− . Cor- recting for the currently accepted value of the Hubble constant, the mass-to-light ratio discrepancy is found to be smaller by a factor of 8.3 [38], but still cannot be explained without the dark matter hypothesis. Similarly, modern galactic mass estimates for galaxy clusters come from the virial theorem and observations of line-of-sight velocity distributions. All of these observations conﬁrm that the ma- jority of the mass in these clusters is not in the luminous component [41,42].

Gravitational Lensing Gravitational lensing oﬀers a unique way of probing mass in the universe, as gravity is coupled to and interacts with mass alone. Gravitational lensing occurs when a massive object in the line of sight to a source bends light from it according to GR, causing distortion or multiplicity of the image of the background source. It can be divided into three main regimes, strong lensing, weak lensing and microlensing.

Strong lensing describes the situation where the lensing object is rather massive. Therefore, we see multiple images of the background source in the case of strong lensing. These multiple images form a so-called Einstein ring, with a 1.4. EVIDENCE FOR DARK MATTER 33 characteristic radius (Einstein Radius) that is proportional to the square root of the mass of the lensing object. Therefore, strong lensing is a way to probe large mass structures.

Unfortunately, strong lenses are quite rare and most lines of sight through the Universe do not pass through a strong lens. In this case, the light deflection is small, only distorting the background. This is quantified by a 2 by 2 matrix that models magnification, shear and rotation. Effectively, a “sheared” image of a spherical object only introduces a 2% distortion to a ellipse. Therefore, in practice, it is required to average over a large number of images to acquire a signal- to-noise ratio of unity in a shear measurement [43]. This is called weak lensing. Similarly, with panoramic cameras now available, lines of sight to millions of stars can be monitored. Therefore, any object passing through these lines of sight can brighten up a star over timescales of weeks or months. This is microlensing.

Bullet Cluster As mentioned before, in galaxies, most of the luminous matter is concentrated in the centre. The dark matter, on the other hand, is mostly found in haloes surrounding the luminous matter. Gravitational lensing can probe these haloes at radii far larger than any visible tracer mass, which has lead to widespread agreement of these structures [43]. The Bullet cluster is one of the prime examples of gravitational lensing providing unique evidence for the existence of dark matter. It consists of two colliding clusters. The different constituents behave differently during the collision, allowing separate study of each component. Stars, traced by visible light, underwent gravitational drag, but were otherwise unaffected. The hot gas, traced by X-ray observations, was more affected than the stars, since the colliding objects interact electromagnet- ically, slowing the hot gas more than the stars. However, this hot gas traces 34 CHAPTER 1. INTRODUCTION most of the baryonic matter. From gravitational lensing data of background sources, it was found that the lensed images trace regions separated from the hot gas (see Fig(1.5)), which can be explained by the dark matter hypothesis rather easily [44].

Figure 1.5: On the left panel, the contours map the matter density distribution using gravitational lensing data from the Magellan Telescope. The right panel shows the same contours overplotted on Chandra X-ray data showing that most of the matter is separated from the hot gas [45].

1.4.3 Cosmological Evidence

Cosmological evidence for dark matter can be found in studies of structure formation and the CMB. Matter perturbations are divided into a dark matter component and a baryonic component and these two components affect the CMB and structure formation differently. Dark matter perturbations are not affected by radiation, but interact purely via gravity. As a result, dark matter collapses to form haloes much faster than any baryonic structure formation, forming potential wells into which baryonic matter can then fall into [23]. Therefore, structure formation is fundamentally driven by dark matter perturbations. [23].

The CMB represents one of the most perfect black bodies we can measure. However, the CMB black body does have anisotropies. These were first measured 1.4. EVIDENCE FOR DARK MATTER 35 by the Cosmic Background Explorer (COBE) [46,47], although the resolution of the experiment was too coarse to describe them in any detail. However, the suc- ceeding CMB experiments like the Wilkinson probe [48,49] and Planck [21] have mapped out these anisotropies with exquisite precision. These anisotropies can be decomposed into an angular power spectrum of temperature anisotropies [see Fig.(1.6)]. This spectrum can be reproduced with the help of computer codes like CMBFAST [50], CAMB [51] and CLASS [52] for a given set of cosmological parameters. The baryons and dark matter affect these peaks differently, respectively. It is quite difficult to accurately reproduce the spectrum without assuming a dark matter component contributing to the energy-density of the Universe. The baryonic component affects the radiation density at recombination while the dark matter perturbations affect the spectrum purely via their gravitational potential. Furthermore, the behaviour of cold dark matter is always non-relativistic, while relativistic effects have to be taken into account while studying the baryonic component and its effects on the CMB. 36 CHAPTER 1. INTRODUCTION

Figure 1.6: The temperature angular power spectrum from Planck data [53].

We see that there is multidimensional evidence for dark matter. However, we still don’t know how dark matter ﬁts into the Standard Model of particle physics, since we do not know what it is actually composed of. Therefore, a bulk of modern research concentrates on looking for new particles that are weakly interacting as extensions to the Standard Model that can explain the dark matter of the Universe. 1.5. DARK MATTER CANDIDATES 37

1.5 Dark Matter Candidates

There are a number of dark matter candidates both within the Standard Model and in extensions of it [54]. This is a growing and exciting ﬁeld of modern research as it represents an overlap of cosmology, the study of the very large scales in the Universe and particle physics, the study of the inﬁnitesimally small. The central property of these particles is that they are all weakly interacting, and they all have mass, through which they manifest as dark matter.

1.5.1 Weakly Interacting Massive Particles (WIMPS)

WIMPS or WIMP-like particles are particles predicted in certain extensions of the Standard Model. It can be shown that a minimally supersymmetric extension (where Bosons and Fermions are related) of the standard model obeys a symmetry called R-parity of the form

R 3(B L)+2S (−1) = (−1) − , (1.24) where B is the baryon number, L is the lepton number and S is the spin of the particle. Under this symmetry, all the supersymmetric partners of the Standard Model fields must be produced in pairs and must decay, while the lightest partner field is stable. If the particle that mediates this field is neutral, it is a dark matter candidate, as its mass is in the GeV-TeV range, but, being the product of a supersymmetric partner field, is very weakly interacting [55]. Similarly, WIMP- like particles are produced in other extensions like the Inert Higgs Doublet model [56]. Unfortunately, many experiments that have been looking for such particles have reported negative results. Recent LHC searches have severely constrained 38 CHAPTER 1. INTRODUCTION the WIMP parameter space [57].

Figure 1.7: Constraints on the WIMP parameter space by the PandaX-II collaboration [57].

1.5.2 Primordial Black Holes and Massive Astrophysical

Compact Halo Objects (MACHOs)

An alternative idea is that primordial compact astrophysical objects like neutron stars and black holes could form the dark matter of the universe, since they have 1.5. DARK MATTER CANDIDATES 39 little to no luminous properties. Gravitational microlensing is an appropriate tool to constrain these objects. The MACHO survey was designed to detect gravitational microlensing eﬀects due to these objects through photometric monitoring of millions of stars in the Large Magellanic cloud. The results seemed to suggest

7 that compact objects of low mass between 10− M ≤ 0.02 M could form up to

20% of the dark matter halo mass [58]. It is also interesting to note that for the smallest primordial black holes, Hawking radiation is an important mechanism for energy loss. Subsequently, the EROS-2 (Exp´eriencepour la Recherche d’Objets Sombres) survey was designed to test the MACHO dark matter hypothesis, which

7 ruled out such objects in the mass range 10− M < M < 15 M [59]. This result suggested that MACHOS cannot make up all of the dark matter currently observed in the Universe.

Current research on the hypothesis that Primordial Black Holes (PBH) could make up the dark matter of the Universe concentrates on ﬁnding a mass range that is not in tension with the current microlensing constraints on such objects. Spectral distortion studies of the CMB have severely constrained these objects under the assumption that they form from inhomogeneities in the early universe [60]. The Advanced LIGO collaboration has also constrained Primordial Black Holes, using their measurement of the stochastic gravitational wave background under the assumption that gravitational wave events could occur due to mergers of PBH [61]. See Fig. (1.8) for the limits on the parameter space from diﬀerent experiments. Recent studies have examined the possibility of spread-out mass function of these objects [62]. 40 CHAPTER 1. INTRODUCTION

Figure 1.8: Limits on Primordial Black Holes from various experiments published by the Advanced LIGO collaboration [61]. The a-LIGO constraints are obtained from the non detection of the stochastic gravitational wave background by O1. These are compared to the constraints from microlensing of stars (red) [63] and quasars (green) [64], CMB distortion measurements (yellow) [60] and dynamical measurements of dwarf galaxies [65]. These constraints establish that f = ΩPBH/Ωm cannot be much larger than 0.1.

1.5.3 Axions

Axions are particles that were hypothesised by Roberto Peccei and Helen Quinn as an extension to the standard model to solve the strong CP problem (see Sec- tion 2.1)[66,67]. Briefly speaking, the axion is a very weakly interacting particle, which could function as dark matter given a non-thermal production mechanism. The axion and axion-like-particles (ALPs) come under the wider classification of Weakly Interacting Sub-eV Particles (WISPS). From a detection point of view, 1.5. DARK MATTER CANDIDATES 41 axions couple to electromagnetic waves, resulting in spontaneous decay to photons, or enhanced decay to photons in the presence of a magnetic field. There have been a variety of laboratory experiments that have attempted detection by concentrating on the axion coupling to electromagnetic waves, but none have had the required sensitivity to truly probe interesting regions of the axion parameter space.

In this thesis, I will give a brief introduction to axions and the constraints on the parameter space from the literature. Then, I will discuss an alternative method of detection of axions using radio telescopes, concentrating on the axion-photon coupling in the context of astrophysical magnetic ﬁelds.

Chapter 2

Axion Models and Constraints

43 44 CHAPTER 2. AXION MODELS AND CONSTRAINTS

2.1 What are axions?

The fundamental motivation for the axion hypothesis comes from the strong CP problem in Quantum Chromodynamics (QCD) [67, 68]. According to QCD, particle-antiparticle symmetry (C-symmetry or Charge conjugation symmetry) and symmetry under reﬂection (P-symmetry or Parity) is violated by particles under the purview of the strong nuclear force. In other words, there is an extra term in the QCD Lagrangian, [69]

g2 L = L + θ¯ GαµνG˜ , (2.1) QCD PERT 32π2 αµν

αµν where LPERT is the full QCD perturbative Lagrangian, G is the gluon ﬁeld ˜ ¯ strength tensor and Gαµν its dual, g is the gauge coupling, θ = θ +Arg [det (M)] where M is the quark mass matrix. The θ-term violates CP symmetry. Any non-zero value of θ would result in a non-zero electric dipole moment for the neutron, dn. However, this has not been corroborated by experiments and there

26 are very strong experimental bounds on dn, of the order of dn < 10− e cm, which

10 corresponds to |θ| < 10− [70]. This was, and still is an unsolved problem in the otherwise generally established QCD. Why does θ, or equivalently, dn have such a value so close to zero? In other words, why does CP appear to be conserved in QCD? This is called the strong CP problem.

An elegant solution to the strong CP problem was provided by Roberto Peccei and Helen Quinn, called the Peccei-Quinn (PQ) mechanism [66, 71]. In the PQ mechanism, θ is essentially promoted to be a ﬁeld. This is accomplished by adding a new global symmetry called the Peccei Quinn symmetry or PQ symmetry, that is spontaneously broken. This results in a new particle called the axion. The 2.1. WHAT ARE AXIONS? 45

KSVZ (E/N=8) 10-6 DFSZ (E/N=0)

10-8

10-10 -1

/GeV -12 γγ

a 10 g

10-14

10-16

10-18 10-8 10-6 10-4 10-2 100 102 ma/eV

Figure 2.1: The axion parameter space showing the axion coupling constant to photons gaγγ versus the axion mass ma. The various constraints on this parameter space will be discussed and exclusion regions will be added through the chapter. See ﬁg. (2.12) for the plot with all the excluded regions with explanations as to where they come from. axion was ﬁrst proposed by Weinberg and Wilczek [72,73], but it was realised very quickly that the range of masses and coupling constants was severely restricted by accelerator experiments [69, 74]. However, models with more weakly interacting axions have since been hypothesised, with the two most popular models being the KSVZ [75,76] and the DFSZ models [77,78]. These models have then established a rather large parameter space for axions to look for, covering many orders of magnitude [see Fig. (2.1)]. 46 CHAPTER 2. AXION MODELS AND CONSTRAINTS

2.2 Properties and models of axions

The PQ symmetry is broken at a scale fPQ, called the Peccei-Quinn energy scale. The mass of the axion, or equivalently, the Peccei-Quinn energy scale is a free parameter of any axion model, the other being the coupling constant, usually gaγγ (the photon coupling), although some more recent studies also consider the coupling to nucleons as well. All masses and coupling constants solve the strong CP problem equivalently well. A variety of cosmological and astrophysical arguments place constraints on the axion mass and its coupling to various other particles. All this will be discussed later in this chapter. Formally, the axion mass is related to the Peccei-Quinn energy scale in the following way: [69,79]

√ 12 z fπmπ 10 GeV ma = ⇒ ma ' 6 µeV . (2.2) 1 + z fPQ/N fa

In the above equation, z ≡ mu/md ' 0.56 where mu ' 5 MeV is the mass of the up quark and md ' 9 MeV is the mass of the down quark, mπ = 135 MeV and fπ = 93 MeV are the pion mass and decay constant, respectively, N is the colour anomaly of the PQ symmetry and fa ≡ fPQ/N.

The eﬀective interaction Lagrangian for the axion with ordinary matter (nucleons, electrons and photons) is [69]

g 0 g 0 L = i aNN ∂ a(N γµγ N) + i aee ∂ a(e γµγ e) + g a E · B . (2.3) int 2mN µ 5 2me µ 5 aγγ 2.2. PROPERTIES AND MODELS OF AXIONS 47

The axion couplings are given by [69]

2 Xe 3α EPQ me gaee = + ln(fPQ)/me) − 1.95 ln(ΛQCD/me) , (2.4) N 4π N fa α gaγγ = (EPQ/N − 1.95) , (2.5) 2π fa X X g = (−F + F ) u − 0.32 + (−F − F ) d − 0.18 aNN A0 A3 2N A0 A3 2N m × N , (2.6) fa

where gaγγ, gaee, gann and gapp are the axion coupling constants to photons, electrons, neutrons and protons, respectively, EPQ is the electromagnetic anomaly of the PQ symmetry, α the ﬁne structure constant, FA0 and FA3 are the axial vector pion-nucleon couplings, me and mN are the masses of the electron and neutron, respectively. From the gaNN term in the Lagrangian, one can derive the coupling to protons gapp and neutrons gann [69]. The main coupling we are concerned with is the axion-photon coupling [see Fig. (2.2)], which quantiﬁes the spontaneous decay of axions into two gamma-ray photons, with a life-time [69] (m /eV) 5 τ = 6.8 × 1024 s a − . (2.7) a 2 (EPQ/N − 1.95/0.72)

Since this coupling is very weak, we are mostly interested in the axion coupling with a magnetic field, decaying into a single photon, which is stronger. This is called the Primakoff effect. 48 CHAPTER 2. AXION MODELS AND CONSTRAINTS

Figure 2.2: This Feynman diagram shows how axions can be converted into γ- ray photons, and vice-versa, via the generic gaγγ coupling and via the Primakoﬀ eﬀect [80].

The two popular models generally quoted by the community are the DFSZ and KSVZ models [75–78]. In the KSVZ model, the axion mass is reduced from the original Weinberg-Wilczek axion, simultaneously adding quarks and scalars to the Peccei-Quinn model. In the DFSZ model, the axion mass is decreased by adding a scalar ﬁeld to the Peccei-Quinn model. Both these mechanisms propose a low mass axion. There exist two mechanisms for non-thermal production of axions in the early Universe, the realignment mechanism and string decay [79, 81, 82], making axions an excellent dark matter candidate. Axions can also be produced thermally, via their couplings to gluons and pions in the primordial thermal bath [69]. 2.3. EXPERIMENTAL CONSTRAINTS 49

2.3 Ongoing Experiments and Experimental Con-

straints

There are several experiments related to or studying axions. A good starting point to describe all of them is Sikivie’s landmark paper in 1983, where he introduced the axion haloscope and helioscope [74]. Sikivie was the ﬁrst to point out that axions have a tendency to cluster about the halo of our galaxy or near the Sun. This led to the many axion dark matter searches that we have today. We have three diﬀerent types of experiments that can look for axions: helioscopes, cavity detectors (haloscopes) and purely laboratory based experiments.

2.3.1 Axion Helioscopes

Axions can be produced in the interior of the Sun by the Primakoﬀ conversion of plasma photons into axions via the inverse of the gaγγ coupling [83]. This gives

gaγγ 11 2 1 rise to a solar flux at the Earth’s surface of φa = 10−10 GeV−1 × 10 cm− s− . In principle, with a strong enough magnetic field, one could convert these solar axions into photons via the Primakoff effect, for a direct detection. The probability that an axion going through a transverse magnetic field will convert to a photon is given by [83]

B 2 L 2 P = 2.6 × 10 17 (g × 1010 GeV)2 F, (2.8) aγ − 10 T 10 m aγγ where B is the magnetic ﬁeld intensity of the magnets used, L is the length of the magnets and the form factor F accounts for the coherence of the conversion process. In other words, F describes how eﬀective the conversion is for a given 50 CHAPTER 2. AXION MODELS AND CONSTRAINTS

axion mode and length of cavity

2(1 − cos(qL)) F = , (2.9) (qL)2

where q is the momentum transfer. Since axions have mass, this results in a loss of coherence between the axions and the photons. The coherence is preserved (F ' 1) for axion masses up to 10 meV, for solar axion energies and magnet lengths of around 10m. For higher masses, F tends to decrease. To mitigate this loss of coherence, a buﬀer gas is introduced into the magnet beam pipes to impart an eﬀective mass to the photons mγ = ωpl (in natural units), where

ωpl is the plasma frequency of the gas. For axion masses equalling the eﬀective photon mass, the coherence is restored. Thus, by controlling the pressure of the gas introduced into the pipes, the sensitivity of the experiment can be extended to higher axion masses as well [83]. 2.3. EXPERIMENTAL CONSTRAINTS 51

Figure 2.3: The CAST Helioscope at CERN [84]

The CERN Axion Solar Telescope or CAST, is an axion helioscope currently in use. In other words, CAST is meant to detect axions produced in the Sun via the Primakoff effect. CAST is a third generation telescope, utilising a de- commissioned test magnet of the LHC of length 9.65 m, capable of producing a magnetic field of up to 9.5 T. This is mounted on a platform with ± 8 degrees of vertical movement, allowing the telescope to observe the Sun for 1.5 hours during both Sunrise and Sunset. At other times, axion background levels are measured. The horizontal movement of ± 40 degrees allows the telescope to follow the Sun’s azimuthal movement throughout the year [84]. CAST has put a limit on the

10 1 axion-photon coupling from the Sun, gaγγ ≤ 0.66 × 10− GeV− , for the axion

2 mass ma < 10− eV [85] [see Fig. (2.4)]. 52 CHAPTER 2. AXION MODELS AND CONSTRAINTS

Figure 2.4: CAST exclusion plot which deﬁnes the excluded values of gaγγ for a range of axion masses [85]. This is one of the most stringent bounds on axion parameters today.

2.3.2 Haloscope Experiments

Sikivie showed that axions in the galactic halo should be converted back into photons in a monochromatic microwave signal in a high-Q microwave cavity, permeated by a strong magnetic ﬁeld [74, 83]. The conversion power is given by 2 ρa 2 P = ηgaγγ B0 VCQL , (2.10) ma where ρa is the local axion density, B0 the magnetic ﬁeld strength, V the volume of the cavity, C a mode dependent form factor, QL the quality factor with the power coupled out to the receiver, η the fraction of the power coupled out by the antenna probe, generally adjusted to be at or near the critical coupling, η = 1/2. The resonant conversion condition is that the frequency of the cavity must equal

2 1 2 v 3 the mass of the axion, hν = mac 1 + 2 %(β ) , with β = c ≈ 10− where v is the 2.3. EXPERIMENTAL CONSTRAINTS 53 galactic virial velocity and c is the speed of light.

(a) The sensitivity of the ADMX experiment, (b) Image of the resonant (c) A schematic of the which goes down to yoc- microwave cavity used in electronics involved in sig- towatt scales [86, 87] ADMX [88]. nal processing of ADMX Figure 2.5: The ADMX experiment

The Axion Dark Matter eXperiment (ADMX) is a haloscope experiment currently running, designed to detect the very weak conversion of dark matter axions into microwave photons, hosted by the University of Washington. The resonant cavity is a circular cylinder, 1 metre long and of 0.5 metre diameter. It consists of an 8 T magnet, and is cooled to 4.2 K by a liquid helium refrigerator to min- imise thermal noise. The experiment makes use of an ultra-low noise microwave receiver, which consists of a cryogenic Superconducting QUantum Interference Device (SQUID) ampliﬁer, followed by ultra-low noise cryogenic HFET ampli- ﬁers. The microwave signal is downconverted and analysed. The ADMX receiver is sensitive to about 0.01 yoctowatts of power, which is equivalent to less than one axion decay within the cavity per minute [86]. ADMX has ruled out KSVZ axions between the mass range 1.91 µeV to 3.53 µeV [89]. 54 CHAPTER 2. AXION MODELS AND CONSTRAINTS

Figure 2.6: ADMX Limit- In this plot, ADMX excludes dark matter axions of a certain mass range, which translates to frequency from E = mc2 = hν, as shown. The mass range corresponding to the frequency bin in the window is 1.9 µeV ≤ ma ≤ 3.53 µeV. This limit is based on the assumption that the local dark matter density is 0.45 GeV/cm3 [89].

The ADMX HF is the High Frequency (HF) variant of the Axion Dark Matter eXperiment, hosted by Yale University. It consists of a 9 T superconducting solenoid, and a dilution refrigerator, that cools the cavity to a temperature of about 100 mK. The copper cavity used has a cavity quality factor Qc of about 20,000, tunable from 3.5 to about 5.85 GHz. The amplification is done with a Josephson Parametric Amplifier, which is basically an LC circuit, with a nonlinear SQUID inductance [90]. This experiment will help to probe axion masses of cosmological significance.

Most of the experiments concerning axion searches exploit the axion coupling to the electromagnetic field. However, there are experiments that attempt to 2.3. EXPERIMENTAL CONSTRAINTS 55 directly detect the axion via its coupling to nucleons and gluons. These couplings result in oscillatory pseudo-magnetic interactions with the dark matter axion field. This field then oscillates at the Compton frequency of the axion, which is proportional to its mass. The Cosmic Axion Spin Precession Experiment (CASPEr) is a haloscope that hopes to detect the Nuclear Magnetic Resonance (NMR) signal of this coupling [91]. It consists of two groups, CASPEr-Wind and CASPEr-electric, that are sensitive to the pseudo-magnetic axion-nucleon coupling and the axion-gluon coupling respectively. They both measure axion spin precession.

CASPEr-Wind is based on the interaction between nucleons and axions. The experiment aims to exploit the interaction between the nuclear spins and the spatial gradient of the scalar axion ﬁeld, as the Earth moves through the galactic axion ﬁeld. The Hamiltonian of this interaction, is given by [91]

p HaNN = gaNN 2ρDM cos(mat) ~v · ~σN , (2.11)

where ma is the axion mass, ~σN is the nuclear spin operator, ~v is the Earth’s velocity relative to the galactic axions, gaNN is the axion-nucleon coupling constant, and ρDM is the local dark matter density, which is determined by astrophysical measurements. The axion mass can be expressed in terms of a frequency, which is more relevant in an NMR discussion. In natural units, the Compton axion frequency is ωa = ma. With the fact that Eqn. (2.11) is the inner product of an oscillating vector ﬁeld and nuclear spins, it can be written as [91]

~ HaNN = γBa · ~σN , (2.12) 56 CHAPTER 2. AXION MODELS AND CONSTRAINTS

where γ is the gyromagnetic ratio of the nuclear spin. As nuclear spins travel through the dark matter halo of the galaxy with a velocity ~v, they can be treated ~ as if they were an oscillating magnetic field Ba of frequency ωa, oriented along ~v. B~ is given by [91] a √ 2ρ B~ = g DM cos(ω t)~v, (2.13) a aNN γ a where gaNN is the axion-nucleon coupling constant and ρDM is the local dark matter density, which is determined by astrophysical measurements. This results in a two-parameter space to explore, the coupling constant and the axion frequency, which is equivalent to the mass, analogous to the electromagnetic experiments. The actual experiment involves the following procedure. First, the leading mag- ~ netic field B0 is introduced to a collection of nuclear spins. This causes these spins to orientate along this leading field, magnetising the axis of the leading ~ field. Then a transverse oscillating magnetic field, Bxy(t) is introduced, oriented ~ in the transverse plane. If the Larmor frequency γB0 is equal to the oscillating field frequency, a resonance occurs, which builds up a transverse magnetisation ~ component. This transverse magnetisation precesses around B0 at the Larmor frequency. This results in a time-dependent magnetic field, that can be picked up by magnetometers, producing an NMR signal. The oscillating transverse ~ magnetic field Bxy is analogous to the oscillating magnetic wind of the axions, ~ Ba.

Similar to CASPEr-Wind, CASPEr-Electric also probes a two parameter space of the axion-gluon coupling and axion Compton-frequency/mass. This rises from

an oscillating nucleon dipole moment induced by the axion-gluon coupling, dn, given by [91] √ 2ρDM dn = gd cos(ωat) , (2.14) ωa 2.3. EXPERIMENTAL CONSTRAINTS 57

2 where gd is the strength of the axion-gluon coupling in GeV− . In this case, a static electric field is applied perpendicularly to the leading magnetic field. The leading field is tuned for resonance. If the resonance condition is met, the axion oscillating dipole moment oscillates at the Larmor frequency. The interaction between this oscillating field and the applied static electric field creates a non- zero oscillating transverse magnetisation, which precesses around the leading field, producing an NMR signal.

2.3.3 Laboratory based experiments

(a) The dimensions of the (b) Schematic of the microwave cavities used in CROWS experiment with the CROWS experiment the two Fabry-Perot [92]. Cavities. Figure 2.7: The CROWS experiment. Fig. (2.7b) shows the two Fabry-Perot cavities whose dimensions are shown in Fig. (2.7a). The one on the left is where axions are produced, while the one on the right is where axions are converted back into photons. The shielding prevents photons from entering the cavity in which axions are reconverted back to photons [92].

There are a number of purely laboratory based experiments looking for axions, in that they do not rely on astrophysical sources for detection. The experiments of the largest scale are about photon regeneration, or “Light Shining through Walls” (LSW) experiments. In these experiments, the idea is that the axions are produced in an emitter cavity. Since axions interact very weakly with matter, they easily penetrate the cavity walls, as well as the barrier between the emitter cavity and another cavity where they will be reconverted back to photons. The 58 CHAPTER 2. AXION MODELS AND CONSTRAINTS detector will aim to detect these photons, that are otherwise unexplained by the Standard Model. The most important experiments are the magnetic birefringence search, PVLAS (Polarizzazione del Vuoto con LASer) and the LSW experiments, OSQAR (Optical Search for QED Vacuum Birefringence, Axion-photon Regen- eration), Any Light Particle Search (ALPS) and the CERN Resonant Weakly Interacting sub-eV Particle Search (CROWS). The results of these experiments will be discussed in this section [92].

PVLAS is an experiment that looks for magnetic birefringence of the vacuum. This phenomenon was predicted a long time ago in Quantum Electrodynamics (QED) [93]. Photons with a polarisation state parallel to an intense magnetic field will interact with the axion field, since axions are produced when photons traverse an intense magnetic field, via the inverse Primakoff effect. This rotates the plane of polarisation of the photons, which can be measured [94,95]. PVLAS has published limits on the ma − gaγγ plane. These limits are roughly gaγγ .

8 1 4 10− GeV− corresponding to ma < 5 × 10− eV [96].

The LSW concept can be understood easily with the help of this schematic. 2.3. EXPERIMENTAL CONSTRAINTS 59

Figure 2.8: Representation of photon regeneration. The scalar, represented by φ is the axion or an ALP in this context. The diagram on the left represents the axion production in the presence of a magnetic field via Primakoff production. The thick vertical line in the middle is the barrier, that blocks photons, but allows axions/ALPs to pass through. On the right hand side, the axions are converted back into photons via the Primakoff effect [97].

Photons are converted into axions in the presence of intense magnetic ﬁelds. These photons are then blocked by a barrier, which allows axions through as they are very weakly interacting. These axions are then converted back into photons in the magnetic ﬁeld on the other side of the barrier. Thus, if light is detected on the other side, one can infer a direct detection of an axion-like-particle.

ALPS and OSQAR were the two important experiments that placed model- independent constraints on axions. Both these experiments employed lasers as their high-intensity monochromatic source, which passed through a magnetic ﬁeld of around 5 T in the case of the ALPS experiment and 9 T in the case of OSQAR.

7 1 3 The ALPS reported limit is gaγγ . 4×10− GeV− corresponding to ma < 10− eV

8 1 [98]. The OSQAR reported limit is gaγγ < 5.7×10− GeV− for a vanishing axion mass [99]. 60 CHAPTER 2. AXION MODELS AND CONSTRAINTS

Figure 2.9: Limits on ALP parameter space by Laboratory LSW Experiments. These limits are model independent, as the only assumption is that axions/ALPs exist and couple to electromagnetic ﬁelds [100,101].

CROWS is the most recent LSW experiment that makes use of the above mentioned mechanism, as well as Fabry-Perot cavities that increase the path traversed by the photons in the magnetic ﬁeld. The CROWS constraint on the

8 1 axion parameter space is gaγγ < 4.9×10− GeV− corresponding to ma < 7.2 µeV [92]. 2.3. EXPERIMENTAL CONSTRAINTS 61

2.3.4 Critical Analysis

Each experiment discussed is designed to constrain axions of a particular range of masses and coupling constants, making a certain a set of assumptions. The larger the range of masses and coupling constants, the better the experiment. Similarly, fewer the assumptions/model-dependencies, the better the experiment. A comparison of the experiments that probe the axion-photon coupling can be seen in table (2.1). Experiments like CASPEr oﬀer alternative methods of detection by probing the axion-nucleon coupling. These can function independently of photon coupling experiments and therefore oﬀer versatility in the experimental approach to detection. Results from these experiments will therefore prove to be comple- mentary. The next decade will be crucial as experimental sensitivities increase and limits could be put on interesting regions in the parameter space. 62 CHAPTER 2. AXION MODELS AND CONSTRAINTS

CAST ADMX Laboratory Experiments (CROWS, PVLAS etc.)

Capable of probing a Capable of achieving These are model- large range of almost benchmark sensitivities independent searches. all astrophysically rele- down to the KSVZ and However, their sensi- vant masses and cou- DFSZ models, but only tivities are quite low. pling constants of ax- for a marginally small There is a long way ions. Relies on the the- range of axion masses. to go to extend these oretical modelling of ax- The high frequency sensitivities down to ions formed in the sun. variant (ADMX HF) interesting regions of However, lacks the sensi- should be able to ex- the parameter space. tivity to probe down to tend this mass range KSVZ and DFSZ mod- to interesting areas of els. Will be superseded the parameter space in by IAXO. the cold dark matter context. However, relies on the measurement of the local dark matter density.

Table 2.1: A critical comparison of the various experiments that probe the axion- photon coupling. 2.4. THEORETICAL CONSTRAINTS 63

2.4 Theoretical Constraints on axion parame-

ters

This section will discuss in detail results from all the diﬀerent searches for axions. Since axions are yet to be discovered, this will consist of the constraints in the axion parameter space, typically in the gaγγ − ma plane. To start with, let us discuss the constraints from astrophysical arguments.

2.4.1 Astrophysical constraints

Solar Constraints There are a number of astrophysical sources that constrain the axion parameters. The closest is the Sun. There are several ways by which axions can be emitted from the cores of stars and cool them. The dominant processes are the Primakoff effect and Bremsstrahlung. In the Primakoff effect, the axions are formed by the interactions between photons and the Coulomb field of the plasma core in stars. In Bremsstrahlung, a plasmon (a quantum of plasma oscillation) is absorbed by an electron or ion, and an axion is emitted, which occurs via the axion-fermion coupling. Therefore, the axion-photon coupling, the axion-electron coupling, as well as the axion mass can be constrained by astrophysical arguments. The effect of axion emission on stars, including the Sun, would be to accelerate their evolution and thus, shorten their lifespans. For an axion mass greater than 1 eV, the axion would carry away energy from the stellar core faster than nuclear reactions would generate it [69]. A simple bound on the axion mass and the axion-photon coupling can be obtained by imposing that the axion luminosity from the Sun must be lower than the surface photon luminosity. If this were not the case, the Sun would have left the main sequence 64 CHAPTER 2. AXION MODELS AND CONSTRAINTS before its current observed age [102]. From this, one obtains a limit on the axion-

6 1 photon coupling, which is gaγγ < 5 × 10− GeV− , for axion masses ranging from 1-1.5 meV [103].

A limit can also be extracted by considering the eﬀect of axion emission on

9 1 the sound speed proﬁle in helioseismological models, gaγγ < 1 × 10− GeV− [104]. The axion cooling of the Sun would require a greater rate of nuclear fuel combustion, thus leading to a greater solar neutrino ﬂux than the measured value,

6 2 1 10 1 4.94×10 cm− s− . This allows an additional limit, gaγγ < 5×10− GeV− [103]. The CAST experiment is the most recent experiment that places a limit on the axion-photon coupling from the Sun [85]. This constraint is based on the non detection of axions by CAST over the axion parameter space deﬁned in ﬁgure (2.4).

Constraints from Globular Clusters The more stringent astrophysical bounds come from observations of globular clusters. Axion cooling via the Primakoﬀ effect also causes the extension of the red giant branch in the Hertzsprung-Russel (HR) diagram, further than is observed. This is because axion cooling would increase the time scale of the helium ﬂash, allowing stars to burn hydrogen in their shells with a degenerate helium core. From this, the axion-electron coupling (as Bremsstrahlung is the dominant emission mechanism in degenerate helium

13 1 cores) is limited to gaee < 3 × 10− GeV− [102]. Axion cooling would shorten the helium burning lifetime of stars. Thus, stars would move away from the Hor- izontal Branch (HB) much quicker than they do in models without axion cooling, resulting in fewer stars in the HB. This reduces the ratio of the number of stars in the HB to the number of red giants in clusters than the observed value today. This is the most stringent astrophysical bound on the axion-photon coupling, for 2.4. THEORETICAL CONSTRAINTS 65

10 1 higher axion masses with gaγγ < 0.66 × 10− GeV− [105].

White Dwarf Cooling and SN 1987A White dwarf cooling is understood with the help of two mechanisms, neutrino emission and surface photon emission, with the latter happening much later than the former [106]. The observation of these objects has revealed expected cooling rates. This allows a limit to be set

13 1 on the axion-electron interaction, gaee < 1.3 × 10− GeV− [103]. However, the most restrictive limit on the axion mass comes from the observation of a rather remarkable event, the cooling of the type II supernova (core collapse) SN 1987A. This supernova resulted from the explosion of the blue super-giant star Sanduleak -69 202 [69]. The energy liberated from the gravitational collapse of the star was of the order of 2 × 1054 erg. This energy was radiated out by thermal neutrinos, whose mean free path was much larger than the size of the core, which was about 10 km, and thus the neutrinos would diﬀuse out. From the hydrodynamic bounce of the Fe core and the gravitational collapse together, a neutrino burst of about 10 seconds is expected. This was conﬁrmed by the detection of 19 anti-electron neutrinos by the KII and IMB neutrino detectors [69,107,108]. This placed severe constraints on the axion cooling, as any coupling that reduced the neutrino burst time period is then ruled out. This allowed a constraint on the Peccei-Quinn

8 energy scale fPQ & 10 GeV, and consequently, a mass constraint on the axion of ma . 10 meV [69,103]. The astrophysical constraints on the axion parameter space can bee seen in Fig. (2.10) 66 CHAPTER 2. AXION MODELS AND CONSTRAINTS

KSVZ (E/N=8) 10-6 DFSZ (E/N=0)

10-8

10-10 -1

/GeV -12 γγ

a 10 g

10-14

10-16

10-18 10-8 10-6 10-4 10-2 100 102 ma/eV

Figure 2.10: The constraints from the axion cooling of astrophysical objects on axion parameter space. The excluded region is shaded in cyan.

It is clear that these arguments constrain axions of higher masses even to coupling constant values below the CAST limit.

2.4.2 Constraints from Cosmology

We know that according to the ΛCDM model, most of the dark matter of the Universe is cold, i.e., non-relativistic in nature. Therefore, for any dark matter candidate to be produced in the early Universe, we require non-thermal production mechanisms. There are two such mechanisms for axions, namely the realignment mechanism and string decay. 2.4. THEORETICAL CONSTRAINTS 67

Realignment Mechanism

In the realignment mechanism, the axion field’s initial value is not at a minimum, causing it to slide down to the minimum and oscillating about it, dissipating energy and decaying until the minimum is attained. The physical basis for the realignment mechanism is that the axion field at early times is basically constant, with the axion being massless. As the temperature of the thermal bath decreases, approaching the QCD transition temperature, the axion begins to gain mass, and rolls down its potential well. At the QCD transition temperature, the axion rolls down the potential well, overshoots the minimum and begins to oscillate. These oscillations produce an axion population that produces the dark matter density today. Once the oscillations die out, the axion comoving number density is conserved [81]. For the misalignment mechanism, an important assumption made is that the misalignment happens before the end of inflation. Otherwise, the dominant mode of axion production is from axion cosmic strings [81,82,109].

If the initial misalignment happened during inﬂation, the axion production is enhanced. Making this assumption, realignment becomes the dominant production mechanism of axions. This results in an axion abundance corresponding to (assuming most of the axions are produced before the QCD transition temperature is reached) [81]

1.19 2 0.41 2 fa Ω h = 0.54g− θ , (2.15) a ,R a 12 ∗ 10 GeV

where g ,R is the number of degrees of freedom of the axion at the time of realign- ∗ ment, θa is the initial angle of misalignment and fa is the Peccei-Quinn energy. From Planck observations [21], we know that the dark matter density parameter, 68 CHAPTER 2. AXION MODELS AND CONSTRAINTS

2 ΩCDMh is about 0.12. By hypothesising that axions are the cold dark matter of

2 2 the Universe, we set Ωa = ΩCDM. By putting Ωah = ΩCDMh = 0.1198 [21] and with the help of Eqn. (2.2) we have

1.19 0.41 2 6µeV 0.1198 ≥ 0.54g−,R θa . (2.16) ∗ ma

From the uncertainty in the value of g ,R, we can extract a mass range for the axion ∗ 2 cold dark matter, assuming θa = π /3. This mass range is 20 µeV ≤ ma ≤ 50 µeV [81].

Axion Cosmic Strings

If the Universe passes normally through the spontaneous breaking of the Peccei- Quinn Symmetry, a global Brownian network of axion strings are formed by the Kibble mechanism [82,109]. As the Universe expands, the strings start oscillating, producing axions. Axionic cosmic strings and their decay after inﬂation is the dominant production mechanism for cold dark matter axions if inﬂation occurred before realignment. This network of strings begins to vibrate and decay with the expansion of the Universe, producing an axion abundance that gives the dark matter density we observe today.

We have two contributions to axion abundance from string decay, loop formation and decay and long string decay. The loop contribution is greater than the contribution from the long strings [81,82,109]. The fundamental parameter that

α is used to quantify the axion production from cosmic string loops is κ , where α is the rate of loop formation and κ is the rate of loop decay or rate of back- reaction. We make the same assumptions as we did for the realignment case, i.e., 2.4. THEORETICAL CONSTRAINTS 69

2 2 ΩCDMh = Ωah = 0.1198 and use Eqn. (2.2). Thus, we have for the contribution from the loops [82],

. α3/2 α 3/2 T 3 f 1 18 Ω h2 = 10.7 1 − 1 + − 0 a , a,l κ κ 2.7 K 1012GeV . α3/2 α 3/2 T 3 6 µeV1 18 = 10.7 1 − 1 + − 0 , (2.17) κ κ 2.7 K ma

where T0 is the CMB temperature. For the contribution from long strings, we have [109]

3 1.18 2 T0 fa Ωa, h = ∆ , ∞ 2.7 K 1012GeV T 3 6 µeV1.18 = ∆ 0 , (2.18) 2.7 K ma where 1/3 < ∆ < 3 is the theoretical uncertainty [82]. The uncertainty in

α α Eqn. (2.17) is in κ . Assuming κ = 0.5 ± 0.2 [81], a CMB temperature of 2.725 K[53] and allowing for the maximum uncertainty in Eqn. (2.18), we can extract a mass range as predicted by the string mechanism for axion cold dark matter production. This mass range is 37 µeV ≤ ma ≤ 1 meV [82]. If the Peccei-Quinn symmetry was broken before inﬂation, the contribution due to topological defects would be negligible, so string decay is overwhelmed by misalignment [110].

Assuming the axion density as a limiting quantity in calculating axion abundance from the misalignment mechanism and string decay, we can extract an axion mass window relevant to cold dark matter, 20 µeV ≤ ma ≤ 1 meV [81,82,110]. This window takes into account various scenarios in the history of the Universe. In other words, it absorbs diﬀerent sequences of events in the early Universe. 70 CHAPTER 2. AXION MODELS AND CONSTRAINTS

Thermal Production Axions are unavoidably produced in the thermal bath of the early Universe. These axions are hot axions, in other words, they are particles produced via thermal processes. Most importantly, these axions undergo relativistic decoupling from the thermal bath, contributing to the hot dark matter density like neutrinos [79, 111]. The most important reactions that produce thermal axions are the axion-gluon coupling and the pion-axion coupling shown in ﬁgure (2.11).

Figure 2.11: The Feynman diagrams for the axion couplings to gluons (left) and pions (right) [110].

Thermal axions contribute to the hot dark matter component of the Universe. The largest contribution to this component is from neutrinos. Thus, the effective neutrino species and mass constrain the population of thermal relic axions. Bounds on these quantities have been set by the recent Planck experiment [21].

Constraints from Planck and BICEP2 The Planck and BICEP2 experiments have placed bounds on a variety of cosmological parameters, with a large amount of precision [21,112]. Since axions decay into photons, they should present a signature on the CMB, which is severely constrained by the above experiments.

2 Thus, assuming an axion density parameter Ωah ≈ 0.12, one can run Einstein- Boltzmann codes, with the usual cosmological parameters and an axion number density from misalignment and string decay as input parameters, and obtain a 2.4. THEORETICAL CONSTRAINTS 71

limit on the axion mass. From this analysis, an axion mass of ma = 76.6 µeV is obtained. By letting the eﬀective mass of the neutrinos become a free parameter, an axion mass of 82.2 µeV is obtained [113]. Another interesting limit, although not relevant to the cold matter scenario, is one on thermal axions, which can also be obtained from Planck data, which is ma < 0.763 eV [114]. This is because, thermal axions contribute to hot dark matter density, change the CMB temperature angular power spectrum, the Sound Horizon as well as details of elemental abundances from Big Bang Nucleosynthesis [114,115]. 72 CHAPTER 2. AXION MODELS AND CONSTRAINTS

2.5 Discussion

The gaγγ − ma plane is being investigated in many different contexts and experiments all around the globe. A summary of the various constraints on the ma − gaγγ plane is shown in figure (2.12). The cavity experiments have ruled out very small regions of axion mass, but they do rely on an assumed local cold dark matter density. The ADMX experiment is currently being upgraded. Along with the ADMX-HF, the cavity detector experiments will be able to explore the axion parameter space relevant for dark matter in the coming decades. The in- surmountable limits are the laboratory constraints, which are model independent. Results from CROWS will further improve on this. However, these experiments lack the sensitivity to robe interesting parts of the parameter space, as of now. The CAST limit 1 is the most recent one, and is the most stringent limit on CDM axions, but also assumes only that axions are produced in the solar plasma. The International Axion Observatory (IAXO) will be the next axion flagship helioscope that will hopefully probe the parameter space down to model sensitivities. The lines of negative slope were obtained by putting in the corresponding values of the decay lifetime of the two-gamma decay. Cosmological arguments help point future axion CDM searches to the most likely regions of detection. The realignment mechanism (yellow hashed region) and string decay (red hashed region) are the main non-thermal production mechanisms for cold dark matter axions (see section 2.4.2). Future searches should be able to probe the regions closer to the model lines in the dark matter mass regime of the axion.

1Thanks to Igor Irastorza for the CAST exclusion data 2.5. DISCUSSION 73

10-6 Laboratory experiments

10-8 Region excluded by Cast and Galaxy cluster measurements 10-10 CAST exclusion -1 KSVZ (E/N=8) DFSZ (E/N=0)

/GeV -12 τγ=Age of Universe ADMX γγ 20

a 10 τγ=10 s g 25 τγ=10 s τ =1030s γ 35 τγ=10 s -14 B=1 nG 10 B=10 nG Thermal Axions 10-16

10-18 10-8 10-6 10-4 10-2 100 102 ma/eV

Figure 2.12: Summary of axion constraints: The lines of positive slope correspond to the KSVZ and DFSZ models of the axion [75–77]. The lines of negative slope correspond to diﬀerent combinations of the axion-photon coupling and axion mass for a particular decay lifetime of the two-gamma decay. The red criss-cross lines indicate the mass range for CDM axions predicted by the string production mechanism. The yellow criss-cross lines indicate the mass range for CDM axions predicted by the realignment mechanism and the navy-blue lines between the two indicate the masses common to both the realignment and the string mechanisms. The CAST [85] limit in blue is one of the most stringent limits on axion parameter space, but is based on solar axion production mechanisms. The Laboratory constraints in dark red are model-independent, and hence insur- mountable [92,96,98,99]. The cyan limit is based on axion cooling and its eﬀects on various astrophysical objects like clusters, stars and supernovae, discussed previously in section 2.4.1. The grey exclusion is based on the ADMX search for 3 axion dark matter, and assumes a local dark matter density of 0.45 GeVcm− [89].

2.5.1 This Project

In this MSc thesis, I describe an alternative method of indirect detection of CDM axions with the help of radio telescopes. The cosmological production mechanisms 74 CHAPTER 2. AXION MODELS AND CONSTRAINTS

establish a mass range for CDM axions which is 1 µeV ≤ ma ≤ 1 meV . Using

2 the relation mac = hν where h is Planck’s constant and ν is the frequency, we can establish the frequency of a photon converted from an axion of mass ma. This frequency range is given by 0.24 GHz ≤ ν ≤ 242.42 GHz. We examine the axion-photon decay in astrophysical magnetic ﬁelds in two main situations, in large scale cosmological objects like galaxy clusters and galaxies and in compact objects, speciﬁcally focusing on neutron stars and pulsars. Eventually, we aim to establish the theoretical framework for a radio observation that could put a limit on CDM axions that is competitive with CAST and ADMX.

Chapter 3

The Axion-Photon Decay in Cosmological Sources

76 3.1. INTRODUCTION 77

3.1 Introduction

It is a well known and an extensively studied fact that the Universe houses magnetic fields of various scales [68], as well as the cosmic background magnetic field [116, 117]. These fields vary in magnitude and scale. Galaxies and clusters of galaxies are home to magnetic fields with magnitudes of tens of µG[118,119]. Since these structures host haloes of dark matter, these systems offer an interesting astrophysical test for the CDM axion scenario. Assuming axions form the dark matter haloes, we could calculate the frequency of the photons resulting from

2 the axion-photon decay in magnetic fields, i.e., the Primakoff effect ν = mac /h, where h is Planck’s constant. Substituting the range of axion masses derived in Chapter2, we obtain a frequency range 2 .8 GHz ≤ ν ≤ 242 GHz, which falls in the radio spectrum. Therefore, radio telescopes could be used to probe CDM axions of all masses.

This chapter is organised as follows. In section 3.2, the relevant quantities will be introduced and the fluxes and intensities observed will be estimated, assuming both spontaneous decay and the Primakoff decay of CDM axions in the presence of galactic and cluster magnetic fields. Section 3.3 will discuss the role of these magnetic fields in this conversion and the significance of their scales. The detectability of these signals are then examined in section 3.4. Section 3.5 concludes the chapter and future directions are discussed. 78 CHAPTER 3. DECAY IN COSMOLOGICAL SOURCES

3.2 Evaluation of Flux and Intensity

These calculations will detail the intensity and flux that would be observed from the axion-photon decay. They are done for both the spontaneous decay and the Primakoff decay. The relevant quantities are first introduced, following which we consider three main scenarios. First, the cosmic axion background will be evaluated, following which the specific intensity of photons from the conversion of axions with column density ρdl in an object at redshift z is evaluated. Finally, the axion-photon flux from various virialised objects at redshift z of mass M is calculated.

3.2.1 Physical Observables and Quantities

The differential line flux density observed is defined as 1

dL Sdf = obs = I dΩ df , (3.1) obs 2 obs obs 4πDc

where S is the ﬂux density, fobs the observed frequency, Lobs the observed lu-

minosity, Dc the comoving coordinate distance to the source, Iobs the observed speciﬁc intensity and Ω the solid angle subtended on the sky. The observed luminosity of an object in the sky is given by

Eobs dLobs = nacdVc , (3.2) τobs

1 dLemit Sdfobs = 2 , where dL is the luminosity distance and dLemit the emitted luminosity 4πdL 2 dEemit dEobs 1 of the object. But, dLemit = = , because of cosmological redshift. Thus, temit tobs 1+z dLobs dLobs dEobs Sdfobs = 2 2 = 2 , where dLobs = . 4πdL(1+z) 4πDc tobs 3.2. EVALUATION OF FLUX AND INTENSITY 79

where Eobs = hfobs is the energy of the observed photons of frequency fobs, τobs is the time taken for the decay in the observer’s frame, nac is the comoving number density of the axions and dVc the comoving volume element. From Rayleigh-Jeans law, we know that

2k I c2 I = B f 2 T −→ T = obs , (3.3) obs 2 obs 2 c 2kBfobs

where T is the axion brightness temperature and kB the Boltzmann’s constant. Since we are considering cold dark matter axions, they are non-relativistic and

2 their energy Ea is given by Ea = mac , where ma is the mass of the axion. We also make note of the Doppler equations

femit dfobs dfemit dv fobs = , τobs = τemit(1 + z) , = = , (3.4) 1 + z fobs femit c

where femit and τemit are the emitted frequency and decay time of the photon, respectively, and z is the redshift at the emitted epoch. From energy conservation, we obtain m c2 E = 2hf −→ f = a . (3.5) a emit emit 2h

We make the same non-relativistic argument in the Primakoﬀ case as well. From energy conservation, one obtains that

m c2 E = m c2 = hf − hf = E − E ⇒ f = a + f , (3.6) a a emit emit emit h where E and f are the energy and the frequency of the magnetic field respectively. For a static field, f=0. Despite cosmological magnetic fields being dynamic in m c2 nature, we assume f to be small as in [120, 121]. Then, f = a . We emit h set Ea = εhfemit where ε is 1 or 2 for the Primakoff and spontaneous decays, 80 CHAPTER 3. DECAY IN COSMOLOGICAL SOURCES respectively. The Compton wavelength of the axion is given by

hc 10 µeV λ = = 0.12 m . (3.7) a 2 2 mac mac

For the two-photon decay, the decay life time is given by [69]

3 g 2 m c2 − τ = 2 × 1039 s aγγ − a . (3.8) γ 10 1 10− GeV− 10 µeV

By Parseval’s theorem, we have

Z Z 1 3 2 1 3 2 ρm = d r|B(r)| = d k|B(k)| , (3.9) µ0V µ0V

where ρm is the magnetic ﬁeld energy density, V the volume of axion conversion and µ0 the magnetic vacuum permeability. We can hence write [121]

3 2 ¯ 2 k B(k) B(ma) ρm(k) = = , (3.10) 2µ0 2µ0

3 2 ¯ 2 where we set k B(k) = B(ma) . From the rate of conversion of axions in a volume V [121] M R ' πg2 a ρ (m ) , (3.11) a aγγ 2 m a ma where Ma is the total mass of axions in the volume V, we can obtain the expression for the time taken for an axion to be converted into a photon in the magnetic

ﬁeld by dividing Eqn. (3.11) by the number of axions N = Ma in the volume a ma V [121]

2 g 2 m c2 B¯(m )− τ = 1.1 × 1031 s aγγ − a a , (3.12) B 10 1 10− GeV− 10 µeV 10 µG 3.2. EVALUATION OF FLUX AND INTENSITY 81

¯ where B(ma) is the magnetic ﬁeld strength on scales of the order of the Compton wavelength of the axion. The ratio of the two is given by

4 2 2 ¯ − τB 9 mac B(ma) = 5.5 × 10− . (3.13) τγ 10 µeV 10 µG

We are interested in the situation where τB 1, since this is the regime where τγ the magnetic field enhances the decay for potential detection. This is true for the dark matter mass range in Fig. (2.12) for magnetic fields of about 10 µG, which are typical in galaxies and galaxy clusters. However, as the magnetic field falls, the Primakoff decay becomes weaker. At magnetic fields of 1 nG wich is the observed value for the primordial magnetic field [116], the spontaneous decay is comparable to the Primakoff decay. It is worth noting that τB = τγ when ¯ B(ma) = Bcritical ' 0.73 nG , for an axion mass of 10 µeV. We now evaluate our three scenarios.

3.2.2 Intensity and Flux

In all our calculations, we convert from the observed frame of reference to the emitted frame according to Eqn. (3.4). We now evaluate the background intensity of axions and corresponding brightness temperature. We substitute Eqn. (3.2) and Eqn. (3.6) in Eqn. (3.1) and express the axion background density in terms

femit of the critical density and use the relation dfobs = − (1+z)2 dz to obtain

hc3 1 3H Ω 1 1 I = ε × 0 a,0 , (3.14) obs 2 4π mac 8πG τemit E(z) 82 CHAPTER 3. DECAY IN COSMOLOGICAL SOURCES

where Ωa,0 = Ωm,0 = 0.3 is the dark matter density parameter today, H0 =

1 1 70 km s− Mpc− the value of the Hubble constant today. This simpliﬁes to

24 7 1 10 µeV 10 s 1 I = 9.5 × 10 Jy sr− × ε . (3.15) back 2 mac τemit E(z)

In this and all future calculations, our ﬁducial values for the axion mass and

10 1 coupling constant are 10 µeV and 10− GeV− , respectively. The typical value for the axion decay time quoted in the literature is 1024 s [69], which we use as our ﬁducial value for the axion decay. This value can be obtained by substituting

a value of 1 eV for the axion mass in Eqn. (3.8). By substituting Iback as the intensity in Eqn. (3.3), we get a temperature of

10 µeV3 1024s (1 + z)2 T ' 5.3 × 102 K × ε3 . (3.16) back 2 mac τemit E(z)

To evaluate the intensity of the axion background from the spontaneous decay, we substitute Eqn. (3.8) in Eqn. (3.15) and ﬁnd

2 2 2 2γ 8 1 mac gaγγ 1 I = 9.5 × 10− Jy sr− , (3.17) back 10 1 10 µeV 10− GeV− E(z) which corresponds to a brightness temperature of

2 2 2γ 12 gaγγ (1 + z) T = 2.7 × 10− K . (3.18) back 10 1 10− GeV− E(z)

12 Therefore, an axion background of temperature ≈ 10− K results from the spon-

10 1 taneous decay of axions of a coupling constant to photons of 10− GeV− . The

magnitude of the magnetic ﬁeld of the universe is approximately Bcritical, which makes the Primakoﬀ axion cosmological background indistinguishable from the 2-photon case [116]. 3.2. EVALUATION OF FLUX AND INTENSITY 83

Now we will evaluate the intensity measured from the axion-photon conversion in a column of some axion dark matter density ρacdlc with some velocity dispersion dv at a redshift z. To start with, consider

2 hfobs hfobs DadlcdΩ dLobs = ε × nacdVc = ε × nac , (3.19) τobs τemit (1 + z)

where Da is the comoving angular diameter distance. This simpliﬁes to

1 5 1 ρacdlc 10 µeV 100 km s− I = 8.9 × 10 MJy sr− × ε obs kgm 2 m c2 dv − a (3.20) 1024s 1 , τemit (1 + z)

1 assuming a ﬁducial virial velocity of 100 km s− , which is typical of galaxies like the Milky Way. From Eqn. (3.3), we get a temperature of

ρ dl 10 µeV3 100 km s 1 1024s 1 T ' 5 × 106 K × ε3 ac c − . 2 2 kgm− mac dv τemit (1 + z) (3.21) To estimate such an intensity and brightness temperature from the spontaneous decay of axions, we substitute Eqn. (3.8) in Eqn. (3.20)

2 2 2 2γ 10 1 mac gaγγ I = 8.9 × 10− MJy sr− × obs 10 µeV 10 10 GeV 1 − − (3.22) ρ dl 100 km s 1 1 ac c − , 2 kgm− dv (1 + z) which corresponds to a brightness temperature of

2 1 2γ 8 gaγγ ρacdlc 100 km s− T = 2 × 10− K (1 + z) . (3.23) obs 10 1 2 10− GeV− kgm− dv

Note that it is possible to recover the background intensity and brightness temperature from Eqns.(3.22) and (3.23) (in the 2-photon case), Eqns.(3.24) and 84 CHAPTER 3. DECAY IN COSMOLOGICAL SOURCES

(3.25) (in the Primakoﬀ case) by substituting [122]

3 dlc (1 + z) ρc = Ωm,0ρcrit,0 , dv H0E(z)

2 3H0 where Ωm,0 = 0.3 is the matter density parameter today, ρcrit,0 = 8πG is the

critical density of the Universe today and H0 is the value of the Hubble constant today.

Now, to estimate the same intensity and temperature for the Primakoﬀ decay of axions, we substitute Eqn. (3.12) into Eqn. (3.20) and obtain

2 2 2 ¯ B 2 1 10 µeV gaγγ B(ma) I = 8.1 × 10− MJy sr− × obs m c2 10 10 GeV 1 10 µG a − − (3.24) ρ dl 100 km s 1 1 ac c − , 2 kgm− dv (1 + z)

which corresponds to a brightness temperature of

ρ dl 10 µeV4 100 km s 1 T B = 0.45 K ac c − × obs 2 2 kgm− mac dv 2 (3.25) g 2 B¯(m ) aγγ a (1 + z) . 10 1 10− GeV− 10 µG

A comparison of the spontaneous decay can be made by calculating the ratio

B 2γ 8 of the two brightness temperatures, Tobs/Tobs ≈ 10 . Thus, it is clear that the Primakoff effect is the dominant mechanism in this case, for magnetic fields of the order of 10 µG.

Next, we aim to estimate the axion-photon ﬂux of an object of mass M at a

comoving distance Dc from us, at a given redshift z. By substituting Eqn. (3.2) and Eqn. (3.6) in Eqn. (3.1) and integrating over frequency, we get for an object 3.2. EVALUATION OF FLUX AND INTENSITY 85 at a given redshift

Z M 1024s 100 kpc2 1 2 Sdf = 1.5 × 1015 Jy Hz . obs 10 10 M τemit Dc 1 + z

(3.26) The ﬁducial distance and mass are 100 kpc and 1010M respectively, which are typical for dwarf galaxies. This ﬂux from the spontaneous decay can be estimated by substituting Eqn. (3.8) in Eqn. (3.26)

Z M 100 kpc2 g 2 Sdf = 0.75 Jy Hz aγγ × obs 10 10 1 10 M Dc 10− GeV−

3 2 (3.27) 10 µeV− 1 . 2 mac 1 + z

We can obtain a ﬂux density from the previous integrated ﬂux by dividing by the bandwidth in frequency space corresponding to the velocity width dv of the object

1 in question according to Eqn. (3.4). We use a fiducial value of dv = 20 km s− as this is typical for objects of our fiducial mass. This corresponds to a flux density of

2 2 6 M 100 kpc gaγγ S = 9.3 × 10− Jy × obs 10 10 1 10 M Dc 10− GeV−

2 (3.28) 10 µeV− 20 km s 1 1 − . 2 mac dv 1 + z

One can repeat the same procedure for the Primakoﬀ decay as well, by substituting Eqn. (3.12) into Eqn. (3.26)

Z M 100 kpc2 g 2 Sdf = 736.4 MJy Hz aγγ × obs 10 10 1 10 M Dc 10− GeV− (3.29) B¯(m )2 10 µeV 1 2 a . 2 10 µG mac 1 + z 86 CHAPTER 3. DECAY IN COSMOLOGICAL SOURCES

As previously calculated in Eqn. (3.28), this corresponds to a ﬂux density of

2 2 2 ¯ 4 M 100 kpc gaγγ B(ma) S = 8.4 × 10− MJy × obs 10 10 1 10 M Dc 10− GeV− 10 µG

20 km s 1 10 µeV2 1 − . 2 dv mac 1 + z (3.30)

2γ 8 Since Sobs/Sobs ≈ 10 , it clear the in this case as well, the Primakoﬀ decay dominates, assuming magnetic ﬁelds of 10 µG.

Therefore, it appears that the Primakoff decay offers a simple way of probing CDM axions, as can be seen from our estimates of the brightness temperatures and flux densities in Eqns. (3.28) and (3.30). However, these estimates share the ¯ assumption B(ma) = 10 µG. This means that the astrophysical magnetic fields under consideration must contain structure over scales comparable to Compton wavelength of the axion [see Eqn. (3.7)], with an amplitude of 10 µG. This is a non-trivial assumption. In [121], it is shown that the large scale magnetic fields exhibit a suppression in this context. In other words, there is a severe absence of power (in the context of axion-photon conversion) on the relevant scales, i.e., the Compton wavelength of the axion. the This can be seen by taking the Fourier transform of the magnetic field, integrating over the relevant length scales. This behaviour will now be discussed in the next section. 3.3. ASTROPHYSICAL MAGNETIC FIELDS 87

3.3 Astrophysical Magnetic Fields

Cluster magnetic ﬁelds are of the order of a few µG. They are modelled with the power law

n n Bk ∝ k− = (klc)− , (3.31)

where k is the magnetic field wave number, lc the coherence length and n the spectral index, which for Kolmogorov turbulence is 11/3 [123,124]. The magnetic field radial profile is given by [118,123,125]

n(e)η B(r) = B0 × , (3.32) n0

where B0 and n0 are the mean magnetic ﬁeld strength and electron density at the centre of the cluster, respectively; ne is the electron density following a β- proﬁle 3β/2 r2 − n (r) = n 1 + , (3.33) e 0 2 rc where r is the distance from the centre of the cluster and rc the cluster core radius. The parameter η is given by [118]

1 η = (2f − 1)(β − 1/6) , β where f is the slope of the correlation between the Faraday Rotation Measure (FRM) and the X-ray flux of the source [118]. A value of η = 0 corresponds to a constant magnetic field, implying f = 0.5. A steeper slope in the correlation results in a higher η value, which leads to a larger magnetic field. Assuming the β-profile for the electron density, we can obtain the magnetic field power law by performing a Fourier transform of Eqn. (3.32). The resulting magnetic field in Fourier k-space is then matched to Eqn. (3.31). Assuming the ΛCDM 88 CHAPTER 3. DECAY IN COSMOLOGICAL SOURCES

model, the parameters of the β-proﬁle evaluated for the Coma cluster are n0 =

3 3 (3.44 ± 0.04) × 10− cm− , rc = 291 ± 17 kpc and β = 0.75 ± 0.03 [123]. The magnetic ﬁeld Fourier transform for the Coma cluster is given by

Z rvir Z rvir 1 ( ik r) 3 1 2 sin(kr) B = B(r)e − · d r = √ B(r)4πr dr , (3.34) k 3/2 (2π) 0 2π 0 kr

where rvir is the virial radius of the object under consideration. The power spec-

3/2 trum B(ma) = k Bk is plotted in Fig.(3.1).

Magnetic Field Power Spectrum in the Coma Cluster B ma μG

1.×10 -9

5.×10 -10

2.×10 -10 k

1 × 21 × 22 × 22 × 23 5 10 1 10 5 10 1 10 pc ¯ Figure 3.1: The power spectrum of the Coma Cluster shows that the B(ma) is 1 1 negligibly low for 0.75 cm− ≤ k ≤ 75 cm− , assuming η = 1. The slope of the line is -2.

The usual range of values of 0.5 ≤ η ≤ 1 mildly alter the shape of the power spectrum, shifting up the line by a negligible amount for higher values of η. The physical reason for this suppression is that the axion conversion to photons is followed by reconversion to axions, as the photons traverse the large scale magnetic ﬁeld. This strong suppression of the magnetic ﬁeld on the scale of the Compton wavelength of the axion would result in impractically long detection times for any radio observation [121,126]. 3.4. DETECTABILITY 89

3.4 Detectability

In this section, we will examine the detectability of the various quantities we estimated in the previous section. We can use the radiometer equation to examine the detectability of these signals

2 1 Tsys τσ = , (3.35) ∆ν ∆TRMS

where ∆TRMS is the noise temperature, ∆ν is the bandwidth of observation which corresponds to a redshift range for a given frequency of emission, Tsys = 20 K is the system temperature, τσ is the integration time for a 1-σ detection. Fig. (3.2) is a schematic representation of the detection of a radio signal. From the radiometer equation, we obtain an estimate of the brightness temperature sensitivity of the telescope (per unit time) for our ﬁducial system temperature for a bandwidth

inst 2 1/2 of 1 MHz, ∆TRMS = 2 × 10− K s .

For reference, an axion of mass 10 µeV would have a frequency of about 1.2 GHz (in the spontaneous decay case) and 2.4 GHz (in the Primakoff case). One can define the bandwidth given this fiducial mass and the typical velocity width of the observed source in the following way

1 m 2 dv ∆f = ac , obs ε h c (3.36) dv m c2 ≈ 8 MHz a 1 1000 km s− 10 µeV

1 where we set ε = 1 and dv = 1000 km s− . In all the detectability calculations, we use this deﬁnition of the bandwidth unless otherwise speciﬁed. We now examine the detectability of the axion background. The signal to noise ratio is calculated 90 CHAPTER 3. DECAY IN COSMOLOGICAL SOURCES

Signal

N-σ

Noise

Figure 3.2: Radio detection of a signal at N-σ, for which an integration time 2 τN σ = 1/N τσ is required. The signal can be the brightness temperature or − the flux (see section 3.2.2). τσ is the integration time as given in Eqn. (3.35) for a brightness temperature or in Eqns. (3.49a) and (3.49b) for a flux signal (see section 3.4.2 for details of detection of flux from a virialised object of mass M).

as a measure of the detectability of the axion background. We can evaluate the signal-to-noise ratio (SNR) as

R fmax S T dfobs/(fmax − fmin) = fmin , (3.37) N ∆TRMS

femit where fmax = fobs = femit at z = 0 and fmin = fobs = 1+z are the observed frequencies corresponding to a redshift range associated to the emitted frequency

femit. For a matter dominated universe, this simpliﬁes to

γ / 2 2 1/2 1 2 S 5 gaγγ τ 10 µeV − = 7.78 × 10− × N 10 10 GeV 1 1 month m c2 − − a (3.38) T 1 sys 1 − (1 + z) 1/2 1 + , 20 K − z

S 2γ where N is the SNR for the 2-photon decay assuming a ﬁducial value of

1 month for the integration time τ and assuming that the bandwidth ∆νobs =

2 mac /2h. 3.4. DETECTABILITY 91

3.4.1 Galactic Centre

We use Eqn. (3.35) to estimate the detection times, where ∆ν is the bandwidth of observation which corresponds to the velocity width dv of the object under observation according to Eqn. (3.4). The axion 2-photon decay intensity from the centre of our galaxy can then be estimated using Eqns. (3.22) and (3.23), assuming

3 2 1 a column density of about 10 M /pc and a velocity width of 100 km s− to be 2 2 2 2γ 9 1 mac gaγγ I = 1.8 × 10− MJy sr− , (3.39) obs 10 1 10 µeV 10− GeV− which corresponds to a temperature of [by substituting the intensity in eqn. (3.3)]

2 2γ 9 gaγγ T = 6.1 × 10− K . (3.40) obs 10 1 10− GeV−

Therefore, the integration time for a 5-σ detection τ5 σ for the axion decay from − a column of cold dark matter at a redshift z is given by

g 4 10 µeV T 2 τ = 4.5 × 106 years aγγ − sys . (3.41) 5 σ 10 1 − 10− GeV− ma 20 K

This is an obviously impractical period of observation time, which makes the spontaneous decay undetectable. The next most obvious alternative is to examine the detectability of the same kind of signal from the Primakoﬀ decay. From the radiometer equation and Eqn. (3.25), in general, the τ5 σ for the detection of − the axion-photon Primakoﬀ decay from a column of cold dark matter axions at 92 CHAPTER 3. DECAY IN COSMOLOGICAL SOURCES a redshift z is given by

7 4 2 10 µeV− g 4 B¯(m )− ρ dl − τ = 0.06 s aγγ − a ac c × 5 σ 2 10 1 2 − mac 10− GeV− 10 µG kgm− 1 2 100 km s 1 − T − sys (1 + z)3 . dv 20 K (3.42)

Similarly, one can estimate τ5 σ for the detection of the Primakoﬀ signal from − the centre of our galaxy assuming a magnetic ﬁeld strength of about 10 µG and a column density of about 103M /pc2

7 2 10 µeV− g 4 T τ = 0.01 s aγγ − sys . (3.43) 5 σ 2 10 1 − mac 10− GeV− 20 K

¯ However, as discussed earlier the magnetic ﬁeld B(ma) 10µG (see section 3.3). Therefore, the detection time evaluated in Eqn. (3.42) increases by several orders of magnitude, making any radio observation impractical. This fact has been overlooked in some of the literature. The estimate of the emission from the galactic centre by Kelley and Quinn [120,127], where they calculate the ﬂux from the galactic centre to be 3.2 µJy assuming that the magnetic at the galactic

1 centre is 50 µG, with a velocity dispersion of 300km s− for an axion mass of

15 1 2.05 µeV and gaγγ = 10− GeV− does not take into account the suppression of the magnetic field. Under the same assumptions and neglecting the severe suppression of the magnetic field, we obtain a flux of 1.4 µJy by integrating over the solid angle subtended by the galactic centre on the sky. 3.4. DETECTABILITY 93

3.4.2 Virialised Objects

We now examine the detectability of the ﬂux density signal from a virialised object according to the equation

2k T 1 2k T 2 ∆S = B sys ⇒ τ = B sys , (3.44) RMS 1/2 Aeﬀ (τ∆ν) ∆ν ∆SRMSAeﬀ

where Aeff is the effective collecting area of the telescope dish. For a telescope like the Lovell Telescope at Jodrell Bank, the collecting area is given by Aeff = 4560 m2. For such a telescope, the flux sensitivity is

2k T ∆Sinst = B sys = 12.1 mJy s1/2 . RMS 1/2 (∆ν) Aeﬀ

Assuming a ﬂux signal as in Eqn. (3.28), we can calculate an integration time for objects at a given redshift. Such a detection can either be resolved, or unresolved. A resolved detection is one where the size of the object on the sky is larger than the telescope beam width. We require that the angular size of the object on the sky be of the size of the angular width of telescope beam, such that the beam covers the entire object’s mass and there is no loss of signal. Unfortunately, nearby dwarf galaxies do no satisfy this condition. To mitigate this, one can set up a system of beams pointing at sections of the object on the sky, where the number of beams is given by

2 2 2 θobj R λ N = 2 = , (3.45) θFWHM Dc Dtel

where θobj is the angular size of the object on the sky, θFWHM is the angular size of the Lovell Telescope like beam at full-width-half-maximum, R the actual size of the object, Dtel the diameter of telescope dish and λ the wavelength of 94 CHAPTER 3. DECAY IN COSMOLOGICAL SOURCES

λ/Dtel

Rob λ/Dtel

Rob

(a) Resolved Detection (b) Unresolved Detection Figure 3.3: The diﬀerence between a resolved and unresolved detection, where θ = λ/Dtel is the telescope beam size and Rob is the size of the object.

observation. We have N = 1 if θFWHM > θobj. The noise ﬂux Sσ increases by a √ √ factor of N for each pixel as given by Sσ = N∆SRMS to obtain

√ 2kBTsys N Sσ = p , (3.46) δνobsτ/N where the integration time τ is the total time taken for the observation. For a 5-σ

1 detection, we have Sσ ≈ 5 Ssignal. For a resolved detection, S > Sσ. Following the same procedure that was used to derive Eqn. 3.26 and dividing by the bandwidth of the signal we obtain the expression for the ﬂux [identical to Eqns. (3.27) and (3.30)] c2M 1 2 S = . (3.47) 2 4πDc ∆νobs 1 + z The bandwidth of the signal is set by the velocity width of the object under

∆v observation as ∆νobs = λ . The velocity width is estimated using the virial 1/3 G1/2 1/2 3 1/3 M theorem ∆v = M 1/2 R . Assuming spherical symmetry, R = 4π ρ . Therefore, we obtain the following expression for the bandwidth

4 1/3 m c2 ∆ν = a G1/2M 1/3ρ1/6 , (3.48) obs 3π ch 3.4. DETECTABILITY 95 which is accurate to a factor of 2, which we choose based on whether the decay is Primakoﬀ or spontaneous. Rearranging Eqn. (3.46) to solve for τ, and substituting for the bandwidth as in Eqn. (3.48) and for N as in Eqn. (3.45), we obtain the expression for the time taken for a resolved detection. Similarly, we obtain the expression for the integration time for an unresolved decay by setting N = 1 in Eqn. (3.46). By substituting Eqn. (3.8) for τemit, we obtain the following expressions for the resolved and unresolved detection respectively for the 2-photon decay , respectively

/ 4 2 1 10 1 3 12 gaγγ − mac − 10 M τ > 1.5 × 10 years 10 10 GeV 1 10 µeV M − − (3.49a) 200ρ 7/6 T 2 1 back sys . ρ 20 K 1 + z

5 4 2 g 4 m c2 − D T τ ≥ 3.36 years aγγ − a c sys 10 10 GeV 1 10 µeV 100 kpc 20 K − − (3.49b) 1/6 10 5/3 ρ 10 M 3 (1 + z) , 200ρback M Thus, for a cluster with, M = 1015 M and at a distance of 100 Mpc we obtain the following integration times for the resolved and unresolved detection respectively

g 4 m c2 4 T 2 1 τ ≈ 8.1 × 1011 years aγγ − a sys , 5 σ 10 1 − 10− GeV− 10 µeV 20 K 1 + z (3.50a) 5 2 g 4 m c2 − T τ ≈ 2.4 × 107 years aγγ − a sys (1 + z)3 , 5 σ 10 1 − 10− GeV− 10 µeV 20 K (3.50b)

Such an estimate is however a little pessimistic, as it assumes that the mass of the virialised object is equally distributed. This is not true, as most of the mass 96 CHAPTER 3. DECAY IN COSMOLOGICAL SOURCES

is concentrated at the centre. An alternate estimate of the integration time for the detection of the decay in a virialised object can be obtained by integrating over the mass covered by the telescope beam assuming a density proﬁle ρ(r), i.e.,

R rbeam 3 M = 0 ρ(r)d r. In this case, rbeam = DcθFWHM, where θFWHM is the beam width of the telescope at half-maximum and Dc is the comoving distance to the virialised object. The advantage of such a mass estimate is that such a detection is always just resolved, i.e., N = 1, which means the integration time can be estimated using Eqn. (3.49b).

The ﬁrst radio telescope search for axions was conducted at the Haystack Ob- servatory at MIT, using their 37 m radio telescope [128]. The authors observed 3 dwarf galaxies from the local group, Pegasus, LGS 3 and Leo I in an eﬀort to

9 1 detect the 2-photon decay. From these observations a limit of gaγγ < 10− GeV−

for axion masses 298 µeV ≤ ma ≤ 363 µeV was inferred [128]. The masses and distances of the dwarf galaxies considered are tabulated below [129].

1 Dwarf Galaxy Distance (kpc) Mass (M ) Velocity Width (km s− )

Leo 1 250 3.3×107 8.8

LGS 3 810 1.3×107 6.5

Pegasus 955 5.5×107 10

Table 3.1: The distances and masses of dwarf galaxies, the latter calculated from their Eqn. (3.51) and the velocity widths provided in [129].

The density-proﬁle of each galaxy is taken to be the spherical isothermal sphere model given by [130] σ2 ρ(r) = LOS , (3.51) 4πG(r2 + a2)

where G is the universal gravitational constant, σLOS the line-of-sight velocity 3.4. DETECTABILITY 97 distribution, r the radius and a the core radius of the dwarf galaxy [128]. The masses were calculated by integrating the density of the galaxy in question over the volume with a cut-off radius of ten times the core radius [128]. By substituting the mass and distance to the respective dwarf galaxies in Eqn. (3.27) and dividing by the velocity distribution of the objects given in table (3.1), a minimum flux can be obtained. By substituting this flux in Eqn. (3.49a), assuming a system temperature of about 200 K for an integration time of about 6 days which are identical to the parameters of Blout et al. [128], we obtain the limit gaγγ <

7 1 3 × 10− GeV− for a 2-σ detection as in [128].

Since we know the size of the dish used in the search and the wavelength of observation (from the target axion mass), we can calculate the size of the beam width at half-maximum θFWHM ≈ 0.35 arcminutes. Therefore, we can estimate the mass covered by the telescope beam, assuming the spherical isothermal model as given in Eqn. (3.51) as

σ2 1 r M(r ) = r − tan 1 beam , (3.52) beam G beam a − a

where rbeam is as deﬁned in section 3.4.2. From such a mass estimate for the

8 1 objects in question in table (3.1), we obtain the limit gaγγ ≤ 3 × 10− GeV− for a a 2 − σ detection. Both of our estimates of the limit are weaker than the reported limit of Blout et al [128] because we support the view that since all the dwarf galaxies have an angular size greater than the Haystack telescope, we need to use Eqn. (3.49a) to determine the integration time for detection, rather than just the regular radiometer equation or use a decreased total mass as estimated using Eqn. (3.52). In the ﬁrst case, the integration time per pointing increases by a factor of N, while the total integration time increases by a factor of N 2. 98 CHAPTER 3. DECAY IN COSMOLOGICAL SOURCES

This leads to the fact that the distance to the object and the eﬀective area of the telescope cancel out exactly, as can be seen from Eqn. (3.49a).

We now consider the detection of the Primakoﬀ ﬂux. Again, we have two cases, for the resolved and the unresolved detections. By substituting Eqn. (3.12) for

τemit and following the same derivation for the integration times, we obtain for the resolved and unresolved case, respectively

7 4 1/3 4 2 ¯ − 10 gaγγ − mac B(ma) 10 M τ > 45 s 10 10 GeV 1 10 µeV 10 µG M − − (3.53a) 200ρ 7/6 T 2 1 back sys , ρ 20 K 1 + z

/ 4 2 5 1 6 2 9 gaγγ − mac − ρ Tsys τ > 10− s 10 1 10− GeV− 10 µeV 200ρback 20 K 4 4 5/3 B¯(m )− D M − a c (1 + z)3 . 10 µG 100 kpc 1010 M

Thus, for a cluster with M = 1015 M and a magnetic ﬁeld of 10µG at a dis-

tance of 100 Mpc we obtain the following integration times for the resolved and unresolved detection, respectively

g 4 m c2 4 T 2 1 τ ≈ 24.37 s aγγ − a sys , (3.54a) 5 σ 10 1 − 10− GeV− 10 µeV 20 K 1 + z

4 2 5 2 4 gaγγ − mac − Tsys 3 τ ≈ 1.2 × 10− s (1 + z) . (3.54b) 5 σ 10 1 − 10− GeV− 10 µeV 20 K

However, as discussed before, the magnetic field in such objects is very low on the scale of the Compton wavelength of the axion, which results in the integration time increasing by several orders of magnitude, making an experimental detection practically impossible. We can quantify this by looking at the power spectrum in fig. (3.1), where we can see that we are overestimated the magnetic field by 3.4. DETECTABILITY 99 around 10 orders of magnitude, which means we underestimated the integration time by about 20 orders of magnitude, which would result in an integration time of about 1011 seconds, which is about 10000 years, even for the most best case

9 scenario integration time of 10− seconds we calculated above. 100 CHAPTER 3. DECAY IN COSMOLOGICAL SOURCES

3.5 Discussion

The possibility of probing CDM axions using radio telescopes is an interesting astrophysical alternative to the particle physics experiments and haloscope detectors in use (see Chapter2). Magnetic fields are ubiquitous in large scale astrophysical objects like galaxies and clusters of galaxies [118, 119]. However, we find that the magnetic fields are not coherent over the scales of the Comp- ton wavelength of the axion [121]. Unfortunately, since the spontaneous decay is quite weak, the detection of any flux from the axion-photon decay in cosmological sources becomes extremely difficult. However, there is another possibility future work could address. Similar to the Sunyaev-Zeldovic effect, the photons from the axion-photon decay scatters off a free electron in the galaxy cluster/galaxy, back reaction is prevented, alleviating the suppression. However, since the electron density of galaxies/galaxy clusters is not very high, any spectral line from this effect should be weak. Nevertheless, it would be worth quantifying this effect and estimating its detection time.

The intensity of the spontaneous decay in the galactic centre is not detectable, as the decay is too weak. The Primakoff decay appears to be a promising detection candidate. However, the Primakoff intensity suffers because of the nature of the large scale magnetic field at the centre of the galaxy. Since other such regions of high column densities are located in other galactic centres much further away than our own, the detection of such an intensity would require an exceptionally high resolution radio telescope, which is currently not achievable. Therefore, we conclude that this regime is not suited to the current technology in radio astronomy. 3.5. DISCUSSION 101

The ﬁrst radio telescope study of the spontaneous axion-photon decay in virialised objects [128] was analysed in section 3.4.2. As stated before, the limit obtained in this study using the same conditions is three orders of magnitude greater

9 1 than the value quoted in [128]. To obtain the same limit, i.e., gaγγ < 10− GeV− , an observation time of about 1.16×109 years would be required for a 1−σ detection, since the angular size of the dwarf galaxies observed on the sky were larger than the beam size of the telescope used in the experiment. For more recent studies, see [126,131].

Therefore, it is clear that we need to focus on more massive objects. Consider the nearby galaxy clusters Virgo, Coma and the clusters ACT CL J0102 4915 (El Gordo) and ACT CL J0438 5419 which are both at redshifts > 0.4 [132]. In table (3.2), the angular size on the sky and integration times are calculated for each cluster using θ = R/Dc for the former and Eqns. (3.49a) and (3.49b) for the latter depending on whether the angular size θ is greater than the telescope beam

3 size of 3.28 × 10− rad (the beam size of the Lovell telescope) respectively.

Cluster Distance (Mpc) Mass (1015 M ) Virial Radius (Mpc) Angular size θ (rad) Integration time (years)

Virgo 18 1.2 2.6 0.14 3.04×1010

Coma 102 2.7 2.8 0.027 2.3×1010

4 9 El Gordo 2957 2.3 2.6 6.1×10− 5.28 × 10

3 8 ACT CL J0438 5419 1629 3 3.3 1.41×10− 3.18×10

Table 3.2: The angular size of the galaxy cluster and corresponding integration times, the latter calculated from their mass provided in [132–135] and the former using the distance provided in [132, 134, 136]. The Coma and Virgo clusters are resolved, while El Gordo and ACT CL J0438 5419 are unresolved, for a telescope with a diameter of 76.2 m like the Lovell telescope at Jodrell Bank, assuming an axion mass of 10 µeV.

Since a resolved detection takes much longer than an unresolved one, one needs to look for massive objects that are unresolved. Nearby virialised objects of large 102 CHAPTER 3. DECAY IN COSMOLOGICAL SOURCES mass have an angular size on the sky larger than typical telescope beams, forcing a resolved detection. However, unresolved massive clusters again have distances that are too large, which again increases the integration time by several orders of magnitude.

The two regimes of detection, i.e., resolved and unresolved, hinge on the apparent size of an object on the sky as compared to the angular beam size of the telescope. The apparent size of an object is the ratio of the diameter to distance of an object. Therefore, for a given distance, the apparent size can be correlated to the mass since the radius is determined from the mass via the virial theorem.

Therefore, one can determine a critical value for the mass for a given distance where the apparent size of the object on the sky is the same as the angular size of the telescope beam, i.e., when the detection is just resolved. As a result, we see that massive objects closer to us become resolved, because their radii are often large, and massive objects further away are too faint. This can be seen by comparing the ﬂux signal as calculated in Eqn. (3.28) for the Virgo and El Gordo,

SVirgo ≈ 2.02 × 104. Corresponding to this, one can plot the integration time as SElGordo function of mass. We show this in (Fig. 3.4). 3.5. DISCUSSION 103

1016 10 Mpc 100 Mpc 1 Mpc 500 Mpc 1015

1014

1013

1012 /years τ

1011

1010

109

108 4 3 2 1 0 1 2 3 4 5 10− 10− 10− 10− 10 10 10 10 10 10 M/1010M

Figure 3.4: The integration time trends are plotted as function of mass for a given distance to the virialised object. The critical mass where the detection is exactly resolved is determined by equating the two integration times obtained in Eqns. (3.49a) and (3.49b).

The critical value of the mass for a distance of 10 Mpc is 1.83 × 107M , 1.823 ×

10 11 10 M for 100 Mpc and 9.22 × 10 M for 500 Mpc. For Dc ≤ 1 Mpc, all objects of cosmologically relevant mass are always resolved. This means that nearby massive objects are too large and therefore require a large integration time to detect. However, unresolved massive objects far away are too faint for detection.

The other alternative is to estimate the mass covered by the beam nearby massive cluster. One can estimate the maximum radius rbeam covered by a typical 104 CHAPTER 3. DECAY IN COSMOLOGICAL SOURCES

telescope beam, such as the Lovell telescope at Jodrell Bank to be

λ ch r =D obs = 2D (1 + z) beam c D c m c2D tel a tel (3.55) 10 µeV 76.2 m D =0.325 Mpc c (1 + z) , 2 mac Dtel 100 Mpc

where Dtel is the diameter of telescope. We can then estimate the mass covered by the telescope beam assuming the Navarro-Frenk-White (NFW) proﬁle [137]

f(cr/r ) M(r) = M vir (3.56) vir f(c)

where c ≈ 5 [138] is the concentration parameter, rvir the virial radius, Mvir the

x virial mass of the cluster and f(x) = ln(1 + x) + 1+x . For the Virgo cluster, this mass is about 9.5 × 1013 M assuming the viral mass and radius in table 3.2.

From this mass estimate, one may infer an integration time of about 65 days for a 5 − σ detection for an axion mass of 200 µeV from Eqn. (3.49b).

The Square Kilometre Array (SKA) will be one of the most powerful tools to probe the radio Universe. However, the very long baselines in the SKA, which are instrumental in providing the collecting area of 1 km2 are ill-suited to detections of the kind discussed in this chapter, as the the beam width is far too small. This has not been addressed in some of the literature [120,126]. To mitigate this, one can use the SKA as a light bucket, rather as an interferometer. In this case, we point all the individual telescopes at the same target in single dish mode. At its full capabilities, the SKA will consist of around a thousand 15 m dishes. Used in single dish mode, this provides a collecting area of around 176715 m2. We tabulate the the mass covered by the beam and therefore, the integration time for each of the objects in table (3.2) in table (3.3) assuming this collecting area. 3.5. DISCUSSION 105

This shows that this approach to the detection is much more eﬃcient as most of the mass in galaxy clusters is concentrated in the central region. While these

10 1 estimates are for a coupling gaγγ = 10− GeV− , the SKA is sensitive to even lower coupling constants.

f(crLovell) f(crSKA) Cluster rLovell f(c) τ5σ (Lovell) rSKA f(c) τ5σ (SKA) 3 Virgo 2.92×10− Mpc 0.011 65 days 0.015 Mpc 0.057 234 s Coma 0.016 Mpc 0.03 8.9 years 0.084 Mpc 0.27 152 hours

Table 3.3: The 5 − σ integration times, τ5 σ, for both the Lovell telescope as well − as the SKA used as a light bucket are tabulated. rLovell and rSKA are the radii covered by the Lovell and SKA beams, respectively, assuming an axion mass of 200 µeV.

Therefore, from estimating the mass of the virialised object within the telescope beam, we avoid wasting time covering less dense parts of the object. Assuming this mode of detection, the Lovell telescope can already probe CDM axions below

10 1 the CAST limit of gaγγ ≤ 0.66 × 10− GeV− , which is currently the most stringent limit on CDM axions, albeit only at 1-σ conﬁdence. Also, the Lovell telescope is only sensitive to axion masses greater than 200 µeV. The SKA used as a light bucket, however, with a much larger collecting area and larger beam width,

11 will allow us to probe the axion parameter space down to a coupling of 10−

1 GeV− at 5-σ conﬁdence. For larger axion masses, the sensitivity is improved to

12 1 approximately 10− GeV− at 1-σ as shown in Fig. (3.5) assuming the virial mass and radius for Virgo as tabulated in table 3.2 and a months integration time. One must keep in mind that these estimates rely on the assumption that the NFW proﬁle reasonably describes the density proﬁle of the Virgo cluster. 106 CHAPTER 3. DECAY IN COSMOLOGICAL SOURCES

10-9 CAST KSVZ (E/N=8) DFSZ (E/N=0) Lovell SKA 10-10

10-11 -1 /GeV γγ a g 10-12

10-13

10-14 10-10 10-8 10-6 10-4 10-2 100 102 104 ma/eV

Figure 3.5: The 1-σ sensitivity of the Lovell telescope and the SKA at its full capacity from 30 days of Virgo observations is shown. The yellow hashed region is the axion mass range from misalignment and light blue hashed region is the axion mass range from string decay. The navy blue hashed region is the intersection of the two. Both the Lovell as well as the SKA can signiﬁcantly improve on the 10 1 cast limit of gaγγ ≤ 0.66 × 10− GeV− [85] for cosmological CDM axions .

Since cosmological sources are ill-suited for detection, compact astrophysical objects like neutron stars with characteristically large magnetic ﬁelds [139] are perhaps better suited for detection. A number of authors have studied the axion- photon decay in the context of neutron stars [140–143]. The next chapter will deal with this system in detail, concentrating on the diﬀerent assumptions made in the literature and their validity. Chapter 4

The Axion-Photon Decay in Neutron Stars

107 108 CHAPTER 4. THE AXION-PHOTON DECAY IN NEUTRON STARS

4.1 Introduction

The motivation to study the axion-photon decay in the context of neutron star magnetospheres comes from the large magnetic ﬁelds associated with these objects [139, 144]. This results in the considerable decrease of the axion decay

¯ 14 time. This can be seen by substituting B(ma) = 10 G in Eqn (3.12), which results in a decay time of 1100 s for axions of mass 10 µeV assuming a coupling

10 1 of 10− GeV− . However, the gain from this large value of the magnetic ﬁeld in the case of neutron stars is oﬀset by a rather small mass to integrate over when compared to cosmological objects. A resonance predicted to occur in these contexts [140–143,145] could alleviate this problem.

This resonance was first studied in 1988 by G.Raffelt and Stodolsky [140]. After a period of relative calm, interest has recently increased with a number of authors studying the resonant decay of axions to photons in neutron star magnetospheres [141–143, 145]. However, not all of these authors agree, with some reporting different results to others. Therefore, there is a need to reconcile these different results and examine the assumptions of each study. This is the objective of this chapter. In section 4.2, the general axion-photon coupled system is derived making as few assumptions as possible, following which an account of the pulsar magnetosphere model is given in section 4.3. Then, a perturbative formalism is introduced in section 4.4 and the critical analysis of the different papers is carried out in section 4.5. Finally, the chapter is concluded in section 4.6 with a discussion and future directions are suggested. Note that only section 4.3 is in standard units, while the rest of the document is in natural units. 4.2. MODIFICATION OF MAXWELL’S EQUATIONS 109

4.2 Modiﬁcation of Maxwell’s Equations

In this section, we will derive the equations for the axion-photon system. A suitable Lagrangian density for the Maxwell’s equations coupled to axions is given by [140] 1 1 1 L = − F F µν + (∂ a∂µa − m2a2) + aF F˜µν , (4.1) 4 µν 2 µ a 4M µν where Fµν = ∂µAν − ∂νAµ is the electromagnetic tensor with Aµ is the vector ˜µν 1 µνρσ potential and F = 2 Fρσ its dual, ma the axion mass, a the axion ﬁeld and M a coupling constant in units of GeV. This Lagrangian allows us to model the axion-photon propagation in vacuum. However, if one were to model the situation near a neutron star, one must include a source 4-current of the form

Jµ = (ρ, J). Thus, the Lagrangian now becomes

1 1 1 L = − F F µν − A J µ + (∂ a∂µa − m2a2) + aF F˜µν . (4.2) 4 µν µ 2 µ a 4M µν

Varying the action with respect to the axion ﬁeld and the vector potential yields the following ﬁeld equations

1 ∂ F µν − J ν − ∂ (aF˜µν) = 0 , (4.3) µ M µ 1 ∂ ∂µa + m2a − F F˜µν = 0 . (4.4) µ a 4M µν

Note that we use the metric signature (+,-,-,-). These equations, along with

˜µν ∂µF = 0 (4.5) 110 CHAPTER 4. THE AXION-PHOTON DECAY IN NEUTRON STARS result in the following modiﬁed Maxwell’s equations

1 ∇ · E − (B · ∇a) = ρ , (4.6) M 1 ∇ × B − ∂ E + (B∂ a − E × ∇a) = J , (4.7) t M t 1 ∂2 − ∇2 + m2 a = − E · B , (4.8) t a M ∇ · B = 0 , (4.9)

∇ × E + ∂tB = 0 , (4.10) where the axion ﬁeld has the dimensions of mass and M = 1012 GeV [141]. 4.3. GOLDREICH-JULIAN CALCULATION 111

4.3 Goldreich-Julian Calculation

One of the ﬁrst solutions of the neutron star magnetosphere that accounted for a charge distribution sorrounding the neutron star is the Goldreich-Julian solution [146]. In this section, this solution will be explained and the resulting expressions for the charge and current derived.

Charged particles moving within a neutron star will experience a force due to the magnetic ﬁeld at its surface. In this solution, the neutron star is assumed to be a perfect conductor. Therefore, there is an electric ﬁeld within the neutron star given by

Ein = −(Ω × r) × B = 0 , (4.11) where Ω is the angular velocity of the neutron star and r is the position vector of the charged particle a distance r from the centre of the neutron star. Since a neutron star is a highly conducting sphere, any electric ﬁeld within the star must be cancelled out by a rearrangement of the charge on the surface to attain a force-free state. From Eqn. (4.11) we can obtain an order of magnitude estimate of the electric ﬁeld at the surface of the neutron star [139]

10 12 1 E ∼ ΩRB0 ∼ 10 − 10 Vcm− , (4.12)

where R is the radius of the neutron star and B0 is the surface magnetic ﬁeld. This electric ﬁeld for typical pulsar parameters exceeds gravity by around 10 orders of magnitude [139]. Therefore, charged particles are pulled out from the surface of the neutron star and form a plasma density around the star, satisfying 112 CHAPTER 4. THE AXION-PHOTON DECAY IN NEUTRON STARS

Eqn. (4.11). The current due to this charge would be given by

J = (Ω × r)ρ , (4.13) where ρ is the charge density assumed to be constant. Consider Maxwell’s equations in their standard form

ρ ∇ · E = , (4.14) 0 ∇ · B = 0 , (4.15)

∇ × B = µ0 (J + 0∂tE) , (4.16)

∇ × E = −∂tB . (4.17)

Following the standard method of deriving the electromagnetic wave equation from Maxwell’s equations, consider ∇ × (∇ × E) = −∇ × (∂tB) = −∂t(∇ × B) . Substituting for the curl of the magnetic ﬁeld, we obtain the following wave equation for E

2 2 − ∇ E + ∇(∇ · E) = −µ0∂tJ − µ00∂t E . (4.18)

Now, we know that J = vρ = ∂trρ = ∂tP, where P is the polarisation density vector. Therefore, the system can be written as

∇ · D = ρf , (4.19)

∇ · B = 0 , (4.20)

∇ × B = µ0∂tD , (4.21)

∇ × E = ∂tB . (4.22) 4.3. GOLDREICH-JULIAN CALCULATION 113

where D = 0E + P is the electric displacement ﬁeld, which obeys ∇ · D =

ρf (the free charge density). This is because the polarisation density vector is deﬁned to be the quantity whose negative divergence yields the bound charge density, i.e., −∇ · P = ρb. Comparing Eqns. (4.21) and (4.16), we have ∂tD =

0∂tE + J. Thus, we now have the following wave-like equation

2 2 µ0∂t D − ∇ E + ∇(∇ · E) = 0 . (4.23)

This wave equation is equivalent to Eqn. (4.18), where the current term has been absorbed by the introduction of the D-ﬁeld. Therefore, any assumptions made about the nature of the D-ﬁeld amounts to a corresponding assumption about the current and therefore, the charge gradient. This point will be crucial, later on in this chapter.

Following Goldreich and Julian [146], we make the following assumptions

• The time derivatives of the electric and magnetic ﬁeld vanish, i.e., the ﬁelds are static. • The charge density around the neutron star is bound to it, i.e., there is no free charge (∇ · D = ρf = 0).

The charge density is given by

ρpl = ∇ · E = −∇ · [(Ω × r) × B] . (4.24) 114 CHAPTER 4. THE AXION-PHOTON DECAY IN NEUTRON STARS

From the vector calculus identity ∇ · (A × B) = B · (∇ × A) − A · (∇ × B) , this then becomes

ρ = − [B · (∇ × (Ω × r)) − (Ω × r) · (∇ × B)] = − [2(B · Ω) − (Ω × r) · (µ0J)] , 0 (4.25) where ∇ × (Ω × r) = 2Ω is the deﬁnition of vorticity. Since we assumed a static

magnetic and electric ﬁeld, ∇ × B = µ0J . Substituting J = vρ and rearranging for ρ, we get Ω · B ρ = −2 , (4.26) 1 − µ |v|2 0 0

where |v|2 = (Ω × r) · (Ω × r) is the speed of the charged particle. An upper bound to the current is Jmax = −2c(Ω · B) .

One can deﬁne the D-ﬁeld as

D = αβE , (4.27)

where αβ is the dielectric permittivity tensor of the medium. The general form of this tensor for a homogeneous plasma is given by [144]

 2 2 2 2  ωplγω˜ ωplωBω˜ ωplγkxv||ω˜ 1 + h 2 2 2 2 i ih 2 2 2 2 i h 2 2 2 2 i ω (ωB γ ω˜ ω (ωB γ ω˜ ω (ωB γ ω˜  2 − −2 2 2 −   ωplωBω˜ ωplγω˜ ωplωBkxv||  αβ = −ih i 1 + h i −ih i ,  ω2(ω2 γ2ω˜2 ω2(ω2 γ2ω˜2 ω2(ω2 γ2ω˜2   B− B− B−  2 2 2 ω2 γk2 v2  ωplγkxv||ω˜ ωplωBkxv|| ωpl pl x ||  h 2 2 2 2 i ih 2 2 2 2 i 1 − h 3 2 i + h 2 2 2 2 i ω (ωB γ ω˜ ω (ωB γ ω˜ γ ω˜ ω (ωB γ ω˜ − − − (4.28) where v is the velocity along the magnetic ﬁeld (taken to be along the z- ||

direction without loss of generality), ki is the component of the wave vector, 4.3. GOLDREICH-JULIAN CALCULATION 115

2 1/2 v|| − γ = 1 − c2 and the brackets < ... > imply an averaging over the summa- tion of the types of particles [144]. The plasma frequency is given by [141]

e2n 1/2 ω = GJ pl m e 0 (4.29) / / / 3/2 3 2 1 2 1 2 11 1 r0 1 km B0 1 s = 1.5 × 10 s− , 10 km r 1014 G P

where nGJ is the Goldreich and Julian charge number density, r0 the radius of the neutron star, B0 the magnetic ﬁeld magnitude, P the period of the neutron star and r the distance from its centre,ω ˜ = ω − kzv is the angular frequency || proportional to the angular frequency of the neutron star in its own reference frame, ω = eB0 is the cyclotron frequency. This exceeds the frequency of the B mec radio emission observed from the pulsar by several orders of magnitude out to a

3 distance of 10 r0 because of the large magnetic ﬁelds observed in these magnetospheres, assuming the pulsar magnetic ﬁeld is homogeneous [144]. Therefore, the dielectric tensor takes the following rather simple form

  1 0 0     αβ =  0 1 0  . (4.30)   2  ωpl  0 0 1 − h γ3ω˜2 i

In other words, the plasma conductivity is non-zero only along the direction of the external ﬁeld [144]. For a curvilinear magnetic ﬁeld with radius of curvature

ρl, the dielectric permittivity tensor is given by [144]

 ω2 v2 ω2 v  1 − 3 h pl || i 0 −ih pl || i 2 γ3ω˜4ρ2 γ3ω˜3ρ  l l    αβ =  0 1 0  . (4.31)   2 2  ωplv|| ωpl  ih 3 3 i 0 1 − h 3 2 i γ ω˜ ρl γ ω˜ 116 CHAPTER 4. THE AXION-PHOTON DECAY IN NEUTRON STARS

It is important to note that the magnetic ﬁeld of a pulsar is not homogeneous. Therefore, using the homogeneous dielectric tensor in Eqn. (4.30) can only be an approximation to the true physical nature of the D-ﬁeld around the pulsar.

The problem at hand is to identify the assumptions made by diﬀerent authors in the literature in studying the resonant axion-photon decay in the magnetospheres of neutron stars and to develop a general formalism from which the results of each of these authors can be derived. The next section will describe the latter. 4.4. GENERAL FORMALISM 117

4.4 General Formalism

We now switch to natural units for the rest of this document, where ~ = c = kB = 0 = µ0 = 1. Our goal is to develop a general mathematical model to describe the axion-photon interaction in the pulsar magnetoshpheres. To achieve this, we study the axion-photon coupled system as a ﬁrst order perturbative system in a background of the uncoupled pulsar magnetosphere as well as a background ﬂow of axion dark matter. Therefore, we have X = X0 + δX where

X = {E, D, B, ρ, J, a}. Our background D0 and B0 ﬁelds obey the following equations

∇ · D0 = 0 → ∇ · E0 = ρ0 , (4.32)

∇ · B0 = 0 , (4.33)

∇ × B0 = ∂tD0 = J0 + ∂tE0 , (4.34)

∇ × E0 = ∂tB0 , (4.35)

The background axion ﬁeld obeys the vacuum Klein-Gordon equation

2 2 2 (∂t − ∇ + ma)a0 = 0 . (4.36) 118 CHAPTER 4. THE AXION-PHOTON DECAY IN NEUTRON STARS

We express the divergence of the D-ﬁeld perturbation as ∇ · δD = ∇ · δE − δρ. Therefore, we now have the following system

1 ∇ · (D + δD) − (B · ∇a + δB · ∇a + B · ∇δa + δB · ∇δa) = 0 , (4.37) 0 M 0 0 0 0 1 ∇ × (B + δB) + (B ∂ a + δB∂ a + B ∂ δa + δB∂ δa − E × ∇a 0 M 0 t 0 t 0 0 t t 0 0

− δE × ∇a0 − E0 × ∇δa − δE × ∇δa) = ∂t(D0 + δD) , (4.38) 1 ∂2 − ∇2 + m2 (a + δa) = (E · B + δE · B + E · δB + δE · δB) , t a 0 M 0 0 0 0 (4.39)

∇ · (B0 + δB) = 0 , (4.40)

∇ × (E0 + δE) + ∂t(B0 + δB) = 0 . (4.41)

At the zeroth order, we recover our original system (4.32- 4.36) for the back-

ground ﬁelds E0, B0, D0 and a0. At ﬁrst order in the perturbations, therefore ignoring terms that are second order in δ, we have

1 ∇ · δD − (B · ∇a + δB · ∇a + B · ∇δa) = 0 , (4.42) M 0 0 0 0 1 ∇ × δB + (B ∂ a + δB∂ a + B ∂ δa − E × ∇a − M 0 t 0 t 0 0 t 0 0

δE × ∇a0 − E0 × ∇δa) = ∂tδD , (4.43) 1 ∂2 − ∇2 + m2 δa = − (E · B + δE · B + E · δB) , (4.44) t a M 0 0 0 0 ∇ · δB = 0 , (4.45)

∇ × δE + ∂tδB = 0 , (4.46) where the background coupling terms remain, since only the background Maxwell terms cancel out with the current and charge source terms. We obtain the propagation equations for δE by evaluating ∇ × ∇ × δE = ∇ × (−∂tδB). Inverting 4.4. GENERAL FORMALISM 119 the order of the derivatives and substituting for ∇ × δB from Eqn. (4.43), we obtain Eqn. (4.48), the propagation equation for δE. Following a similar procedure for δB, we obtain Eqn. (4.49). Therefore, we have the following propagation equations

1 ∂2 − ∇2 + m2 δa = − (E · B + δE · B + E · δB) , (4.47) t a M 0 0 0 0 1 − ∇2δE + ∇δρ = −∂2δD + ∂ [B ∂ a + δB∂ a + B ∂ δa − E × ∇a − t M t 0 t 0 t 0 0 t 0 0 1 δE × ∇a − δE × ∇a − E × ∇δa] − ∇(B · ∇a + δB · ∇a + B · ∇δa) , 0 0 0 M 0 0 0 0 (4.48) 1 (∂2 − ∇2)δB = − [∇ × (B ∂ a ) + ∇ × (B ∂ δa) + ∇ × (δB∂ a )− t M 0 t 0 0 t t 0

∇ × (E0 × ∇a0) − ∇ × (E0 × ∇δa) − ∇ × (δE × ∇a0)] , (4.49) where as before, the coupling background terms survive, since they do not cancel with the background Maxwell source terms. One can, through a redeﬁnition of the parameters, make the equations dimensionless. It is clear that by dividing Eqn. (4.47) by a factor of M 3 on both sides, the dimensions cancel out. Similarly, by dividing Eqn. (4.48) by a factor of M 4 on both sides cancels out the dimensions.

˜ 1 ˜ 1 ˜ 1 ˜ 1 1 1 By deﬁning E = M 2 E , B = M 2 B , D = M 2 D , J = M 3 J , ρ˜ = M 3 ρ, M δa = δa˜ 120 CHAPTER 4. THE AXION-PHOTON DECAY IN NEUTRON STARS

we have a completely dimensionless system

˜ 2 ˜ 2 2 ˜ ˜ ˜ ˜ ˜ ˜ ∂t − ∇ +m ˜ a δa˜ = −[E0 · B0 + δE · B0 + E0 · δB] , (4.50)

˜ 2 ˜ ˜ ˜2 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ − ∇ δE + ∇δρ˜ = −∂t D + ∂t[B0∂ta˜0 + δB∂ta˜0 + B0∂tδa˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ − E0 × ∇a˜0 − δE × ∇a˜0 − E0 × ∇δa˜] − ∇(δB · ∇a˜0 ˜ ˜ ˜ ˜ ˜ ˜ + B0 · ∇a˜0 + δB · ∇a˜0 + B0 · ∇δa˜) , (4.51)

˜ 2 ˜ 2 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ (∂t − ∇ )δB = −[∇ × (B0∂ta˜0) + ∇ × (B0∂tδa˜) + ∇ × (δB∂ta˜0)− ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ∇ × (E0 × ∇a˜0) − ∇ × (E0 × ∇δa˜) − ∇ × (δE × ∇a˜0)] . (4.52)

˜ ˜ Then,a ˜0 , E0 , B0 , m˜ a are all dimensionless parameters. Our ﬁducial value for M is as in [141].

12 2 30 B0 10 GeV B˜ = 1.95 × 10− × , (4.53) 0 1014 G M 12 2 34 E0 10 GeV E˜ = 2.3 × 10− × , (4.54) 0 12 1 10 Vcm− M 1012 GeV m m˜ = 10 27 × a . (4.55) a − M 1 µeV 4.5. CRITICAL ANALYSIS OF LITERATURE 121

4.5 Critical Analysis of Literature

One of the first attempts at understanding axion-photon mixing in this context was made by G.Raffelt and Stodolsky [140] in 1988, where the axion-photon system was solved in a vacuum background with a strong constant magnetic field. After a long dormant period, there has been in recent times a renewed interest in this subject. Work has been done to try and modify the formalism in [140] to account for the plasma in the pulsar magnetosphere [141–143, 145]. All of these papers have different results. Our goal is to try and reconcile these different results to different assumptions made on the general system derived in the previous section. We start with an account of the system in [140], followed by [141] and [145], obtaining two different solutions to the problem. A comparison of these solutions and discussion on future directions of research concludes this section and chapter.

4.5.1 Raﬀelt and Stodolsky

In [140], the authors solve the axion photon mixing equations assuming that the background fields are in vacuum and the axion scalar field is relativistic. Further- more, the authors assume that the external magnetic field is constant in time, but not necessarily constant in space. This system is obtained by substituting

E0 = a0 = ρ = J = 0 and we set B0 to be constant in time. The authors also note that although ∇ · δE 6= 0, they neglect these terms at the level of the coupling to be negligibly small, i.e. ∇(B · ∇a) → 0. This simpliﬁcation is made based on the assumption that the axion only mixes with the parallel photon state [140]. The 122 CHAPTER 4. THE AXION-PHOTON DECAY IN NEUTRON STARS

following background equations follow

∇ · E0 = 0 , (4.56)

∇ · B0 = 0 , (4.57)

∇ × B0 = ∂tE0 , (4.58)

∇ × E0 = −∂tB0 . (4.59)

(4.60)

The system of equations to solve then becomes

1 ∂2 − ∇2 + m2 δa = − (δE · B ) , (4.61) t a M 0 1 ∂2 − ∇2 δE = B ∂2δa . (4.62) t M 0 t (4.63)

The above system can be derived from Eqn. (4) in [140] by applying the trans-

E formation A = , keeping in mind that the QED parameters Qj = 0 in our || iω system. Taking the Fourier transform in time and assuming the axion only mixes with the parallel component of the electric ﬁeld (taken to be along the y-direction without loss of generality) [140], the axion-photon mixing matrix can be written as      0 iBtω/M δA 2 2 || ω + ∂z +     = 0 , (4.64)   2    −iBtω/M −ma δa

where δA = δEz/iω and Bt = |B0| sin θ, where θ is the angle between the || propagation direction and the external ﬁeld. The authors develop a perturbative solution to account for the inhomegeneity in the magnetic ﬁeld in the “interaction picture” of quantum mechanics [140]. To linearise this equation, one 4.5. CRITICAL ANALYSIS OF LITERATURE 123

approximates ω + i∂z = ω + k ≈ 2ω [140]. Note that this approximation is made assuming intrinsically relativistic axions. One can cast the resulting ﬁrst order equation in the form of a “Schr¨odinger”equation [140]

i∂rA = (H0 + HI )A , (4.65) where   0 0 H0 =   , (4.66)  2  0 −ma/2ω and   0 i∆  B  HI =   , (4.67) −i∆B 0

B where ∆B = 2M sin θ, in the “interaction picture” of quantum mechanics. Con- sider a system where HI is the perturbation Hamiltonian, i.e., we have the following time evolution

i∂t|ψ(t)i = (H0 + HI )|ψ(t)i . (4.68)

Now, to switch to the interaction picture, we transform the state vector in the following way

iH0t |ψ(t)i = e− |φ(t)i . (4.69)

Substituting this back into the time evolution, we get

iH0t iH0t i∂t|φ(t)i = e− (HI )e |φ(t)i = Hint|φ(t)i . (4.70) 124 CHAPTER 4. THE AXION-PHOTON DECAY IN NEUTRON STARS

If we deﬁne Σncn(t)|ni = |φ(t)i, we then have

while the ﬁrst order solution is

Z t (a) Hint(t0) |φ(t) i = dt0 |ψ(0) . (4.73) 0 i

mn mn mn iH0 t iH0 t i(Em En)t mn iωmnt mn Hint = hm|e− HI (t)e |ni = e− − HI (t) = e− HI (t) ,

deﬁnes the Hamiltonian perturbation for a transition from state |mi to state

|ni(of energy Em and En, respectively), we can calculate the probability of this transition as 2 Z t mn iωmnt HI e− Pm n = dt0 . (4.74) → 0 i

We have seen that a formalism is defined that treats the axion-photon resonant mixing in a general way. However, as explained in section 4.3, the magnetosphere contains a charge distribution due to the electric field induced at the surface of the neutron star. Therefore, a modification of this formalism to take into account this charge distribution and the current distribution due to it is necessary. 4.5. CRITICAL ANALYSIS OF LITERATURE 125

4.5.2 Hook et al

In [141], the authors also solve the axion-photon mixing equations in the same perturbative formalism developed in [140], but with a different set of assumptions. They do not assume a vacuum around the neutron star, rather, the background electric and magnetic fields are identical to the Goldreich-Julian solution. There- fore, the mixing equations change from those obtained by [140], because of the effect of the Goldreich-Julian charge and current, D = E + P (J = ∂tP). The axions in question are assumed to be non-relativistic or at most semi-relativistic. Therefore, QED birefringence effects can be neglected [141]. The final assumption is that the cyclotron frequency Ωc is much larger than the plasma frequency ωpl as well as the frequency of the photon ω. This is just an artefact of the assumption that the magnetic field from the pulsar dominates over other terms (therefore E0 terms are neglected). By substituting a0 = const and setting B0 to be constant in time in the D-field [141] regime in Eqns. (4.47) and (4.48), we get

B 1 −∇2δE + ∇δρ = −∂2δD + 0 ∂2δa − ∇(B · ∇δa) , (4.75) t M t M 0 1 (∂2 − ∇2 + m2)δa = − (E · B + δE · B ) . (4.76) t a M 0 0 0

From Eqn. (4.27), we can transform from the D-field to the E-field. The effect of such a transformation on the system depends on the dielectric permittivity tensor αβ. Choosing a particular form is equivalent to setting constraints on the components of ∂tJ and consequently ∇ρ. The physical significance of the dielectric tensor is that its form reveals the conductivity of the plasma along different directions. Therefore, choosing a particular αβ is the same as choosing a particular current that results from it. This allows certain charge and current derivatives to vanish, but not all. For example, substituting for the homogeneous 126 CHAPTER 4. THE AXION-PHOTON DECAY IN NEUTRON STARS

αβ in Eqn. (4.75) gives

1 (∂2 − ∇2)δE = ∂ δρ + ∂2δaB − ∂ (B · ∇δa) , (4.77) t x x M t 0 x 0 1 (∂2 − ∇2)δE = ∂ δρ + ∂2δaB − ∂ (B · ∇δa) , (4.78) t y y M t 0 y 0 ω2 1 ∂2 1 − h pl i − ∇2 δE = ∂ δρ + ∂2δaB − ∂ (B · ∇δa) , (4.79) t γ3ω˜2 z z M t 0 z 0 (4.80)

from which one can clearly see that ∂tJx = ∂tJy = 0, which means that ∂xδρ =

∂yδρ = 0. This means that the plasma conducts only along one direction. There- fore the electric ﬁeld components along the other directions do not propagate and decouple from the equations [141,147]. Thus, following [141], we take the Fourier

2 2 2 transform in time, obtaining ω = k + ma, where k is the wave vector. We set the dielectric tensor to be

  1 0 0

 2   ωpl 2  αβ =  0 1 − sin θ 0  , (4.81)  ω2    0 0 1

where δEx = δEz = ∇δρ = ∂tδJx = ∂tδJz = 0, where the form for the tensor is such since we assume that the plasma conducts only along the z-direction. 1

1 Factorising ω2 gives the following mixing matrix [141] 1 pl cos2 θ − ω2

  ω2 ω2 g Bω2 θ   − pl aγγ sin   ω2 ω2 δEy  1 pl cos2 θ 1 pl cos2 θ  δEy − ∂2   =  − ω2 − ω2  .   , (4.82) z        gaγγ B sin θ 2 2  a ω2 (ω − ma) a 1 pl cos2 θ − ω2 1Since the plasma is assumed to be conducting, ∇δρ need not vanish (See section 4.6). 4.5. CRITICAL ANALYSIS OF LITERATURE 127

1 where gaγγ = M and θ is the angle between the external ﬁeld (taken to be along a transverse direction B sin θ = Bz) and the direction of propagation, under the ω4 pl 2 2 2 2 assumption that | ω2 sin θ cos θ| |ω − ωpl|.

We now switch to a local coordinate system, by setting zˆ = rˆ. The next simpliﬁcations hinge on the assumption that ω remains relatively constant in the thin resonant conversion region around rc. A WKB approximation

d2δA˜ dδA˜ d2δa˜ dδa˜ || k || , k , dr2 dr dr2 dr

iωt ikrθ is applied with plane wave solutions of the form δa(r, t) = ie − a˜(r) , δA (r, t) = || iωt ikrθ δEy e − δA˜ (r) , where δA = in the above mixing equation. This simpliﬁes || || iω the mixing equation to the following

     2 2 ˜ d 1 ma − ξωpl ∆B δA −i +    ||  = 0 , (4.83)  dr 2k     ∆B 0 δa˜

sin2 θ gaγγ Bω sin θ where ξ = ω2 and ∆B = ω2 . We then solve for the conversion 1 pl cos2 θ 1 pl cos2 θ − ω2 − ω2 probability using the formalism of transition amplitudes in the interaction picture of quantum mechanics (see section 4.5.1). Adapting this formalism to our situation, we get [140,141]

R r 2 2 2 Z r 0 d¯r[ma−ξ(¯r)ω (¯r)] B(r0)ξ(r0)gaγγ pl 2mavc paγ = dr0 × e . (4.84) 0 2vc sin θ

We can evaluate the asymptotic value of this integral by taking vc limr paγ = →∞ pa∞γ . This is evaluated by the method of stationary phase. Consider the following integral Z ∞ ikφ(r) I = f(r)e− dr . (4.85) 0 128 CHAPTER 4. THE AXION-PHOTON DECAY IN NEUTRON STARS

If φ(r) has a stationary point at r = r0, then

1 00 φ(r) = φ(r ) + 0 + (r − r )2φ (r ) + O(r − r )3 . (4.86) 0 2 0 0 0

Ignoring third and higher order derivatives, the integral now is

Z 1 2 00 ikφ(r0) ∞ ik (r r0) φ (r0) I = f(r0)e− e− 2 × − dr . (4.87) 0

2 2 1 2 00 One can make the substitution i y = 2 ik × (r − r0) φ (r0) ⇒ q 00 q 00 iπ/4 kφ (r0) iπ/4 kφ (r0) y = e− (r −r0) 2 ⇒ dy = e− 2 dr . Thus the integral evaluates to s 1 2π ikφ(r0)+iπ/4 f(t0)e− 00 . (4.88) 2 φ (r0)k

Applying this formula to Eqn. (4.84), we ﬁnd [141]

2 2 πrcB (rc)gaγγ 1 2 2 2 paγ∞ ≈ = gaγγB(rc) L , (4.89) 3ma 2vc

q where L = 2πrcvc is the distance over which the conversion takes place at 3ma θ = π/2 [141].

dP The power radiated per unit solid angle can then be calculated as dΩ = q rc 2 rc 2 2GMNS 2 × p∞ρ v r , where ρ = ρ∞ is the dark matter density at the a DM c c DM DM √π rc resonant conversion length rc, calculated from the background density ρDM∞ [141]. The ﬂux estimate assuming a distance of 100 pc to the pulsar and an axion mass of 6.6 µeV is given by [141]

1 5 100 pc 6.6 µeV 200 km s− dP/dΩ S = 6.7 × 10− Jy , 2 8 d ma v0 4.5 × 10 W (4.90) 4.5. CRITICAL ANALYSIS OF LITERATURE 129

where v0 is the dark matter background velocity. The bandwidth of this signal is set by the background velocity of the dark matter because any kinetic energy gained by particles entering the gravitational potential well of the neutron star is lost via the gravitational redshift as the particles exit the potential well [141]. This assumption is only valid if one does not take into consideration the coupling of the axions to gravity. In this is not the case and the impact parameter is smaller than the resonant conversion radius, this assumption might be too simplistic, as the path followed the photons will not necessarily be the same as the in-falling axions. Furthermore, the gradients of the ﬁelds appear as ﬁrst order corrections to the trajectory of both the axions and the photons, in which case the one-dimensional treatment in the literature may not be adequate.

As a check of the analytic estimates, one can solve the system of equations that describe the axion-photon mixing Eqn. (4.82). We use the values quoted

14 6 2 in the paper, B0 = 10 G = 1.95 × 10− GeV is the magnetic ﬁeld of the

12 1 neutron star at r0 = 10 km (the radius of the neutron star), gaγγ = 10− GeV− is the coupling constant of axions to photons, θ = π/2 is the angle between the propagation direction and the magnetic field direction, ma = 0.658 µeV which 2 vc corresponds to a frequency of 1 GHz, ω = 1 + 2 ma is the frequency of the axion where vc = 0.11 is the velocity of the axion at the resonant conversion / / 11 r0 3 2 14 r0 3 2 radius rc, ωpl = 1.06 × 10 GHz × r = 6.98 × 10− GeV × r . The solution of this system of differential equations is then plotted as a function of r in Fig. 4.1. The plot shows that the absolute probability of conversion increases to a maximum at r = rc. Note that this not a probability density. It is clear from this plot that probability peaks at the resonant conversion region and is then cut-off. 130 CHAPTER 4. THE AXION-PHOTON DECAY IN NEUTRON STARS

Probability of axion-photon conversion as a function of distance from the neutron star 0.12

0.1

0.08

0.06

0.04

0.02

0 200 300 400 500 600 700 800 900 1000 r/km Figure 4.1: The Probability of conversion as a function of distance from the neutron star. The boundary conditions are set at rinitial = 223 km. The initial conditions are Ey(rinitial) = Ey0 (rinitial) = a0(rinitial) = 0 and a(rinitial) = 1 GeV. 2 Ey The probability of axion photon conversion is plotted as | | . ainitial | |

4.5.3 The Landau-Zener Solution

The two approaches to the problem provided in [140] and [141] diﬀer in their assumptions of the pulsar magnetosphere and in their derivation of the linearised axion-photon mixing matrix. The assumption made by Hook et al [141] that the axions are non-relativistic results in the ﬁnal diagonal component of the axion-

2 2 photon mixing matrix ω − ma to vanish, unlike the mixing matrix obtained in Raffelt and Stodolsky [140]. The second difference is that the D-field introduces the plasma density into the mixing matrix, because the pulsar magnetosphere contains a charge distribution due to the electric field on the surface of the star (see section 4.3). The authors in [145] solve the system introduced by Hook et 4.5. CRITICAL ANALYSIS OF LITERATURE 131 al by making the initial assumptions made by Raffelt and Stodolsky [140] and introducing a similar plasma outside the neutron star as in Hook et al. [141]. They then solve the system using the Landau-Zener formalism. Before detailing their solution, we briefly discuss the Landau-Zener formula and its derivation.

Consider a two level system described by a Hamiltonian of the following form

  − −f   H(x) =   , (4.91) −f where = αx is linearly dependent on x and α, f > 0 are constants. Consider a

Schr¨odinger-like equation of the form (setting ~ = 1)

    ∂ c1(x) c1(x) i   = H(x)   . (4.92) ∂x     c2(x) c2(x)

This leads to the following ordinary diﬀerential equation

d2 c (x) + [f 2 − iα + (αx)2]c (x) = 0 (4.93) dx2 2 2

i π 1/2 for the amplitude c2(x). The substitution x → z(x) = e− 4 (2α) x in the above equation results in the Weber diﬀerential equation,

d2 1 1 c˜ (z) + ν + − z2 c˜ (z) = 0 , (4.94) dz2 2 2 4 2 132 CHAPTER 4. THE AXION-PHOTON DECAY IN NEUTRON STARS

if 2 wherec ˜2(z) = c2(z(x)) and ν = 2α . Assuming the system is initially in the state |1i, we can set initial conditions to be

2 |c1(x → −∞)| = 1 (4.95)

2 |c2(x → −∞)| = 0 (4.96)

The off-diagonal terms in the Hamiltonian result in a population transfer from state |1i to state |2i. The aim is to find the final distribution.

2 2 PLZ = |c1(x → ∞)| = 1 − |c2(x → ∞)| . (4.97)

Among the four solutions to the Weber diﬀerential equation, only the function

D ν 1(−iz(x)) vanishes for x → −∞. Therefore c2(x) fulﬁls the initial condition − − (4.96) if it is written as

c2(x) =c ˜2(z(x)) = AD ν 1(−iz(x)) , (4.98) − −

√ i −πγ where the normalisation constant A = γe− 4 is set by the initial condition

f 2 (4.95), where γ = −iν = 2α . By substituting the asymptotic expressions for

D ν 1 − −

2 −π (ν+1)i i R ν 1 D ν 1(−iz(x → −∞)) = e 4 e− 4 R− − (4.99) − − √ 2 2π π νi i R ν D ν 1(−iz(x → ∞)) = e 4 e 4 R (4.100) − − Γ(ν + 1)

√ back into Eqn. (4.92), where R = 2αx, the Landau-Zener formula can be derived to be

2πγ PLZ = e− . (4.101) 4.5. CRITICAL ANALYSIS OF LITERATURE 133

In [145] the authors have adapted the Landau-Zener formalism to the resonant conversion of axions to photons in the neutron star magnetosphere. We have the same derivation for the mixing matrix plus the extra contribution from the plasma density quantiﬁed by the dielectric tensor [see Eqn. 4.30)], essentially the same initial assumptions as in section 4.5.2. Linearising in the same manner as in [140], we have the following mixing matrix (this result diﬀers from [141] as the axion-photon term in the mixing matrix doesn’t vanish)

    d δa ω + ∆a ∆M     i = 2 , (4.102) dz    ωpl 2  δA ∆M ω − sin θ || 2ω

ω2 2 pl 2 ma B 2 where ∆p = − 2ω sin θ, ∆a = − 2ω and ∆M = 2M sin θ where θ is deﬁned as the angle between the magnetic axis and the direction of propagation. The axion

2 2 2 resonance is deﬁned to occur when the diagonal elements are equal ma = ωpl sin θ.

Introducing the quantity ∆k and the mixing angle θm, we get [145]

2 2 1/2 1 ∆M ∆k = 2[(∆a − ∆p) )/4 + ∆M] , tan(2θm) = . (4.103) 2 ∆a − ∆p

Thus, the propagation equation can be rewritten in the following manner

       d δa ∆ + ∆ ∆k cos 2θm sin 2θm δa i   =  ω + p a I +     , dz    2 2     δA sin 2θm − cos 2θm δA || || (4.104) where I is the 2×2 identity matrix. The eigenvalues and eigenvectors of Eqn. (4.104) are     ∆ + ∆ ∆k δa cos θm k = ω + p a + ,   =   , (4.105) + 2 2     δA sin θm || + 134 CHAPTER 4. THE AXION-PHOTON DECAY IN NEUTRON STARS and

    ∆ + ∆ ∆k δa − sin θm k = ω + p a − ,   =   , (4.106) − 2 2     δA cos θm || − respectively. These ‘+’ and ‘-’ modes represent the photon and the axion states respectively. To calculate the conversion probability, one can express the axion and photon waves in terms of two ‘+’ and ‘-’ amplitudes A+ and A in the − following way [145]

      δa cos θm − sin θm   = A+   + A   . (4.107)     −   δA sin θm cos θm ||

Following [145], substituting this expansion and ignoring the term proportional I in Eqn. (4.104), one gets the equation for the evolution of the amplitudes

      δa ∆k/2 iθ A    m0   +    =     . (4.108) δA −iθm0 −∆k/2 A || −

The evolution equation is now in a form where the Landau-Zener formula is applicable. We can now express the adiabaticity ratio Γ as [145]

∆k/2 Γ = . (4.109) θm0 4.5. CRITICAL ANALYSIS OF LITERATURE 135

The larger the adiabaticity ratio is, the greater the suppression of the adiabatic conversion. The probability of conversion can then be expressed by the Landau-

Zener formula. At the resonant conversion length, ∆a − ∆p = 0. This simpliﬁca- tion results in the expression

∆2 B(r )2 Γ = M = c , (4.110) 0 2 2 ∆p M ma

keeping in mind that ωpl = ma at the resonant conversion length rc ≈ 170 km

10 (from setting ωpl(r) = ma). Substituting M = 10 GeV and taking the derivative of the plasma frequency in Eqn. (4.29), we get the probability of conversion

11 Pconv = 1 − exp(−2πΓ) ≈ 9.8 × 10− . The probability of conversion from axions to photons is given by 1 − PLZ, since the Landau-Zener formula is applied to the non-adiabatic evolution, while the conversion of the parallel mode of the photon to axions and vice-versa is an adiabatic evolution [145]. Therefore, we may now estimate the flux from this conversion. Following a similar derivation for the flux density to that in chapter2, we obtain a flux density directly proportional to the total mass of the axions Ma

M P 1 S = a conv . (4.111) 2 4πDc τemit∆ν 1 + z

To estimate the mass of the axions, we assume the same overdensity near the neutron stars due to the gravitational potential of the star as in [141], from

2 √ which we obtain that ρc = vaρ where the background density ρ = √πv0 ∞ ∞ 3 1 0.45 GeV cm− , the background velocity of the axions v0 ≈ 100 km s− , while we assume the velocity of the axions to be 0.1 as the authors in [145] assume

3 relativistic axions. Therefore, we obtain ρc ≈ 1.07×10 ρ . We can now estimate ∞ 136 CHAPTER 4. THE AXION-PHOTON DECAY IN NEUTRON STARS the total mass of the axions as

4 M = πr3ρ ≈ 9.24 × 10 3 kg . (4.112) 3 c c −

Substituting for Pconv, Eqn. (3.12) for τemit and B(rc) for the magnetic ﬁeld and a distance of 100 pc we obtain

2 2 2 8 100 pc 10 µeV gaγγ S ≈ 6 × 10− µJy × D m 10 10 GeV 1 c a − − (4.113) 100 km s 1 B(r ) 2 − c , dv 2 × 1010 G where the bandwidth ∆ν is set by the background velocity as before. Interest- ingly, this ﬂux is three orders of magnitude lower than the ﬂux obtained by Hook et al [141]. 4.6. DISCUSSION 137

4.6 Discussion

We have developed a formalism where the classical axion-photon mixing is modelled in a rather general way. In our formalism, we assume the contribution from quantum terms such as QED birefringence terms in [140, 145] to be vanishingly small in our regime. This is related to the assumption that axions are ’cold’, hence non relativistic. In [141] the authors do validate this assumption, setting the axion kinetic energy near the neutron star to be equal to the gravitational

2 potential energy of the neutron star, i.e., v = GM/r0 where M is the mass of the neutron star and r0 its radius. From this argument, no more than semi-relativistic axion velocities are inferred. Future work could examine this assumption and see if there is more to the story.

In our perturbative formalism, we set the background to be the cold dark matter axion flow around the pulsar, uncoupled to the pulsar electromagnetic system. At first order in the perturbations, the axion field is coupled to photons, and we compare our general system to the others obtained in the literature. This makes the general assumptions made in the recent literature and the reasons for the varying results clear. We address the different assumptions and critically examine their validity. We also identify areas where these estimates can be made more accurate/realistic.

The authors in [140,145] solve the coupled axion-photon system in the assuming implicitly relativistic axions. While the authors in [140] solve the system in a vacuum background, the authors in [145] adapt their formalism to the pulsar magnetosphere filled with plasma, as derived by Goldreich and Julian [146] and confirmed in simulations [148]. They then obtain the Landau-Zener behavior 138 CHAPTER 4. THE AXION-PHOTON DECAY IN NEUTRON STARS for their solution. This is because the fundamental feature of the Landau-Zener solution is that the difference of the two energy states described by the two-level

Hamiltonian are linearly dependent on time, i.e. E2 − E1 = αt where E1 and E2 are the energies of the two states (of the photon and the axion), α is a constant proportional to the velocity of the perturbation variable of the system, here the axion translation [149]. A large velocity yields a large non-adiabatic transition probability. Therefore, the assumption of relativistic axions should (as it does) yields to Landau-Zener behaviour. However, since axions are assumed to be the dominant component of cold dark matter, one has to determine if the assumption of relativistic axions is realistic. Even if one assumed an enhanced density of axions near the pulsar due to their relativistic velocities, the ﬂux obtained is weaker than that obtained assuming non-relativistic axions (which don’t exhibit Landau-Zener behaviour). This is due to the fact the Landau-Zener formula gives a very low probability for the adiabatic conversion of the axions to photons (see section 4.5.3 and the conversion in this case is assumed to take a place at point and not at a region of space as in [141].

Hook et al [141] assume non-relativistic (or at the most semi-relativistic) axions in their treatment of problem. As a result, the derivative of E2 − E1 = ω − ma is vanishingly small. Since such a situation cannot be modelled by the Landau- Zener formula, the authors derive the conversion probability according to the perturbative formalism in the interaction picture of quantum mechanics developed in [140]. Assuming the the collecting area of the Lovell telescope to be 4560 4.6. DISCUSSION 139 m2, we get the following integration time for a 1-σ detection

/ D 4 6.6 µeV7 3 200 kms 1 7 g 4 τ ≈ 4900 s c − aγγ − × σ 12 1 100 pc ma v0 10− GeV− 4/3 8/3 2 B − P − A − 0 , 1014 G 1 s 4560 m2 (4.114)

Therefore, the CDM axions can be probed well below the CAST limit with just the Lovell telescope. At 1 − σ conﬁdence level, assuming a target pulsar/neutron

14 star at a distance of 500 pc, magnetic ﬁeld of 10 G, mass of 1 Modot, radius of 10 km and period of 1 s, the Lovell telescope is capable probing CDM axions down to model sensitivities (KSVZ and DFSZ axions) for larger axion masses with an integration time of 30 days. The SKA at full capacity has the capabilities to probe almost the whole range of CDM axion masses down to the model sensitivities under the same assumptions. This is shown in Fig. (4.2). 140 CHAPTER 4. THE AXION-PHOTON DECAY IN NEUTRON STARS

10-9 CAST KSVZ (E/N=8) DFSZ (E/N=0) Lovell (8 kpc) 10-10 Lovell (500 pc) Lovell (1 kpc) SKA (500 pc) SKA (1 kpc)

10-11 -1

10-12 /GeV γγ a g

10-13

10-14

10-15 10-10 10-8 10-6 10-4 10-2 100 102 104 ma/eV

Figure 4.2: The 1 − σ sensitivities of the Lovell telescope and the SKA are shown for varying target pulsar distances, assuming a pulsar mass of 1 M , magnetic ﬁeld of 1014 G, a period of 1 and a radius of 10 km. The yellow hashed region is the mass range from misalignment, the light blue hashed region is the range from string decay and the navy blue hashed region is the intersection of the two. It is clear that radio telescopes are excellent ways of probing CDM axions at very competitive couplings.

The WKB approximation used by the authors in [141] hinges on the fact that the axion mass ma = ωpl in the conversion range, resulting in a simple dispersion

2 2 relation k = ωpl−ma. However, such a dispersion relation is clearly not valid away

from the conversion region since plasma frequency ωpl is a function of distance r. As a result, an approximate asymptotic conversion probability is derived. However, a more exact estimate of the conversion probability could be obtained by deriving a gradient expansion to obtain a ﬁrst order equation that is valid across the entire trajectory of the axion. Also, it is very important to consider that the axion passes two critical points in its trajectory, i.e., as it enters and leaves the resonant conversion region, as any line not tangential intersects a circle 4.6. DISCUSSION 141 in two places. This point seems to be missed by Hook et. al. in their numerical estimates.

In literature, the ∇ · E term is assumed to be vanishingly small. In [140], the authors work in the vacuum regime, where this assumption could perhaps (to some extent) be valid. However this does not explain why the coupling term

B0 · ∇a is vanishingly small. The authors do derive formally that the axion only mixes with the component of the electric ﬁeld parallel to the direction of propagation. It remains to be seen if this validates the assumption that ∇ · E is vanishingly small. It is clear that the magnetosphere of neutron stars is not a vacuum [139,146]. Therefore, the authors in [141,145] make a strong assumption in neglecting all components of the charge gradient in the mixing matrix, even if the coupling term vanished from the geometry of the axion mixing. Recent studies have shown that axions introduce a vacuum-bound charge and current in the absence of free charge and current as assumed here [150]. Future work must take into account this gradient, as well as the variation of the dipolar magnetic ﬁeld in the Landau-Zener formalism, neglected by [145].

Finally, the axion background derivatives seem to be neglected in the literature. Future work could examine the validity of this assumption and check to see if a constant axion ﬁeld solution is compatible with the formalism developed here.

Chapter 5

Conclusions

143 144 CHAPTER 5. CONCLUSIONS

The cold dark matter problem remains one of the main unsolved problems in cosmology. There are several cold dark matter candidates (see section 1.5 for WIMPS and PBHs and [54] for a review), of which the axion is a well motivated candidate. With several decades of fruitless searches for WIMPS [57] and new strict constraints on PBHs [61] (see Fig. 1.8), interest in axions has increased in the past decade. Several experiments of various kinds (see section 2.3) exist exploring axion parameter space.

One can exploit the axion-photon coupling to look for the axion-photon decay in astrophysical objects possessing magnetic ﬁelds. These can be classiﬁed into cosmological objects such as galaxies and clusters of galaxies and compact objects such as neutron stars. Axions produced from non-thermal production mechanisms decay into radio-photons, making radio telescopes useful instruments for detection.

Our analysis of the axion-photon decay in cosmological objects can be divided into three main scenarios. We derive an axion cosmic background assuming from the spontaneous decay. We ﬁnd that this is far too weak to detect. Secondly, we derive the intensity from the conversion of a column of axions with a column

density ρcdlc at redshift z Eqns. (3.22) (for the spontaneous decay) and (3.24) (for the Primakoff decay). We find that this intensity in the case of the spontaneous decay is too weak, but would be potentially detectable assuming a magnetic field of 10 µG. 145

In cosmological objects, typical magnetic fields have amplitudes of tens of µG [117, 118]. Even with such low field values, because cosmological masses are typically greater than 1010 M , one could in principle integrate over the whole mass to obtain a detectable signal. However, magnetic fields in such objects are coherent over length scales of parsecs, which makes B(ma), the magnetic field on scales of the Compton wavelength of the axion negligibly small [see Fig. (3.1)] [121]. This effect increases integration times by several orders of magnitude, making any experiment impractical.

While the axion cosmic background from the spontaneous decay is far too weak to detect, virialised objects of large mass could present an detection opportunity. This is our final calculation for the cosmological objects. We derive a flux that would be observed by the conversion of CDM axions in a virialised object of total mass M a comoving distance Dc [see Eqns. (3.28) and (3.30)]. Because of the scale problem with magnetic fields, we concentrate on the spontaneous decay in the context of detection of virialised objects. However, we find that massive objects nearby are too large to detect with a single telescope beam, requiring several beams to cover their whole area. This is because virialised objects of large mass have a large virial radius, and therefore have a larger angular size on the sky than the beam width of the telescope. Therefore, integration times increase to impractical values [see Eqn. (3.49a)]. Therefore, we need an unresolved detection, for which we have to turn to massive objects further away whose angular size is the same as or smaller than the telescope beam. However, massive objects further away are too faint for detection [see figure 3.4] for a more elaborate discussion]. As a result, we disagree with the results of the first telescope search for axions in dwarf galaxies [128] as the telescope beam width of the search was smaller than the angular size of the dwarf galaxies on the sky (see section 3.4.2 for a more 146 CHAPTER 5. CONCLUSIONS elaborate discussion).

An alternate method for the detection of virialised objects is to integrate over the mass covered by the telescope beam. This allows just resolved observations of nearby objects. We present two estimates of this case, with a radio telescope (the Lovell Telescope) and the Square Kilometre Array (SKA) (see section 3.5). The SKA cannot be used as an interferometer in this case, since the longest baselines (which are important for a large collecting area) make the telescope beam rather small. This leads to the high resolution that the SKA aims for, but is undesirable for detections of the kind discussed here of virialised objects. Therefore, for such a detection, the SKA can be used as a light bucket, with all the individual radio telescopes of the array pointing at the same target in single dish mode. In general, we ﬁnd that this method of integrating over the fraction of the total mass covered by the telescope beam is feasible to probe axions of larger masses (ma ≥ 200 µeV) (see table 3.3). In this case, the sensitivities of the Lovell and the SKA from observations of the Virgo cluster are shown in Fig. (3.5) albeit at 1 − σ. Although one can set quite competitive constraints from radio observations of nearby massive clusters, the sensitivities aren’t quite at the level to model the KSVZ and DFSZ axion models at CDM axion masses.

Therefore we conclude that cosmological objects are ill-suited sources for detection in the near future, with telescope searches only sensitive to larger masses and require knowledge of the density profile of the target clusters. With this in mind, however, there is some scope for future work. One could examine the probability of axions converting to photons and then scattering off free electrons in cosmological objects, similar to the Sunyaev Zeldovich effect. Such a phenomenon would prevent back reaction and result in a spectral line in the radio. 147

The axion-photon decay in the magnetospheres of neutron stars has been recently studied extensively, motivated by the resonant conversion in large magnetic fields [140–145]. However, there is some disagreement between authors in the literature. This is due to different assumptions and different kinds of formalism adapted. In this work, an attempt has been made to derive a general formalism that reconciles the different results of authors. Therefore, a general perturbative formalism has been developed, in the context of which the different assumptions have been identified and critically examined [see sections 4.4, 4.5 and 4.5.3]. This allows one to determine the best way to proceed to obtain more accurate estimates and perhaps design a radio telescope experiment for detection.

The authors in [140,145] assume implicitly relativistic axions, because of which they obtain Landau-Zener behaviour in their solutions. On the other hand [141] works with non-relativistic axions, deriving the velocities from the gravitational potential of the neutron stars. Since axions are assumed to be cold dark matter candidates, relativistic velocities are unlikely. We ﬁnd that the ﬂux from such relativistic axions is weaker than the non-relativistic case, despite relativistic velocities enhancing the density of axions near the neutron star. This is because the Landau-Zener formula results in low probabilities of conversion of the axions to photons.

The conversion of non-relativistic CDM axions dealt with in [141] results in a ﬂux signal given by Eqn.(4.90). The detectability of such a signal is examined in section 4.6. From Eqn. (4.114), we conclude that the Lovell telescope is capable

12 1 of probing CDM axions down to a coupling of 10− GeV− , at 5 − σ conﬁdence. The 1 − σ sensitivity of the Lovell telescope probes CDM axions down to model sensitivities for axions of ma ≥ 200 µeV for pulsars at a distance of 500 pc with 148 CHAPTER 5. CONCLUSIONS parameters considered in [141]. With the advent of the SKA, the future of radio astronomy is poised to enter a new era of sensitivity and resolution. The SKA at its full capabilities is sensitive enough to probe most CDM axion masses at model couplings as shown in Fig. (4.2) assuming a months observation of pulsars with parameters discussed in [141] and up to a distance of 500 pc.

While it is well understood that axions can resonantly convert to photons in neutron star magnetospheres, there is scope to refine the estimates in the literature. To accurately determine the resultant flux and signal features, one needs to derive first order expansions that trace the entire trajectory of the axions through the two critical points. This formalism needs to be reconciled to the quantum picture of cross sections derived by Sikivie [74] and used in [120,121,127]. Finally, the ∇(∇ · δE) term that seems to be ignored in the literature needs to be examined, as the presence of charge perturbations δρ make this term non- vanishing. Bibliography

[1] I. Newton. Philosophiae Naturalis Principia Mathematica. Auctore Js. New- ton. 1687.

[2] A. Einstein. Zur Elektrodynamik bewegter K¨orper. Annalen der Physik, 322:891–921, 1905.

[3] H. S. Leavitt. 1777 variables in the Magellanic Clouds. Annals of Harvard College Observatory, 60:87–108.3, 1908.

[4] E. Hubble. A Relation between Distance and Radial Velocity among Extra- Galactic Nebulae. Proceedings of the National Academy of Science, 15:168– 173, March 1929.

[5] A. Einstein. Die Feldgleichungen der Gravitation. Sitzungsberichte der K¨oniglichPreußischen Akademie der Wissenschaften (Berlin), Seite 844- 847., 1915.

[6] A. Friedmann. Uber¨ die Kr¨ummung des Raumes. Zeitschrift fur Physik, 10:377–386, 1922.

[7] G. Lemaˆıtre. Un Univers homogènede masse constante et de rayon crois- sant rendant compte de la vitesse radiale des nébuleusesextra-galactiques. Annales de la SociétéScientifique de Bruxelles, 47:49–59, 1927.

149 150 BIBLIOGRAPHY

[8] H. P. Robertson. Dynamical space-times which contain a conformal Eu- clidean 3-space. Transactions of the American Mathematical Society, Vol. 29, pp. 481-496, 29:481–496, 1927.

[9] A. G. Walker. On Milne’s Theory of World-Structure. Proceedings of the London Mathematical Society, (Series 2) volume 42, p. 90-127, 42:90–127, 1937.

[10] R. A. Alpher, R. Herman, and G. A. Gamow. Thermonuclear Reactions in the Expanding Universe. Physical Review, 74:1198–1199, November 1948.

[11] G. Gamow. The Evolution of the Universe. Nature, 162:680–682, October 1948.

[12] A. A. Penzias and R. W. Wilson. A Measurement of Excess Antenna Tem- perature at 4080 Mc/s. The Astrophysical Journal, 142:419–421, July 1965.

[13] R. H. Dicke, P. J. E. Peebles, P. G. Roll, and D. T. Wilkinson. Cosmic Black-Body Radiation. The Astrophysical Journal, 142:414–419, July 1965.

[14] R. V. Wagoner, W. A. Fowler, and F. Hoyle. On the Synthesis of Elements at Very High Temperatures. The Astrophysical Journal, 148:3, April 1967.

[15] S. W. Hawking and G. F. R. Ellis. The Large Scale Structure of Space-Time. 1975.

[16] S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Castro, S. Deustua, S. Fabbro, A. Goobar, D. E. Groom, I. M. Hook, A. G. Kim, M. Y. Kim, J. C. Lee, N. J. Nunes, R. Pain, C. R. Penny- packer, R. Quimby, C. Lidman, R. S. Ellis, M. Irwin, R. G. McMahon, P. Ruiz-Lapuente, N. Walton, B. Schaefer, B. J. Boyle, A. V. Filippenko, T. Matheson, A. S. Fruchter, N. Panagia, H. J. M. Newberg, W. J. Couch, BIBLIOGRAPHY 151

and T. S. C. Project. Measurements of Ω and Λ from 42 High-Redshift Supernovae. The Astrophysical Journal, 517:565–586, June 1999.

[17] A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan, S. Jha, R. P. Kirshner, B. Lei- bundgut, M. M. Phillips, D. Reiss, B. P. Schmidt, R. A. Schommer, R. C. Smith, J. Spyromilio, C. Stubbs, N. B. Suntzeﬀ, and J. Tonry. Observa- tional Evidence from Supernovae for an Accelerating Universe and a Cos- mological Constant. The Astronomical Journal, 116:1009–1038, September 1998.

[18] F. Zwicky. Die Rotverschiebung von extragalaktischen Nebeln. Helvetica Physica Acta, 6:110–127, 1933.

[19] V. C. Rubin and W. K. Ford, Jr. Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions. The Astrophysical Journal, 159:379, February 1970.

[20] K. Schwarzschild. On the Gravitational Field of a Mass Point According to Einstein’s Theory. Abh. Konigl. Preuss. Akad. Wissenschaften Jahre 1906,92, Berlin,1907, 1916, 1916.

[21] Planck Collaboration XIII., P. A. R. Ade, N. Aghanim, M. Arnaud, M. Ash- down, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, J. G. Bartlett, and et al. Planck 2015 results. XIII. Cosmological parameters. Astronomy and Astrophysics, 594:A13, September 2016.

[22] Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. B. Barreiro, N. Bartolo, S. Basak, R. Battye, K. Benabed, J.-P. Bernard, M. Bersanelli, P. Bielewicz, J. J. Bock, J. R. Bond, J. Borrill, F. R. Bouchet, F. Boulanger, M. Bucher, 152 BIBLIOGRAPHY

C. Burigana, R. C. Butler, E. Calabrese, J.-F. Cardoso, J. Carron, A. Challi- nor, H. C. Chiang, J. Chluba, L. P. L. Colombo, C. Combet, D. Contreras, B. P. Crill, F. Cuttaia, P. de Bernardis, G. de Zotti, J. Delabrouille, J.-M. Delouis, E. Di Valentino, J. M. Diego, O. Doré,M. Douspis, A. Ducout, X. Dupac, S. Dusini, G. Efstathiou, F. Elsner, T. A. Enßlin, H. K. Erik- sen, Y. Fantaye, M. Farhang, J. Fergusson, R. Fernandez-Cobos, F. Finelli, F. Forastieri, M. Frailis, E. Franceschi, A. Frolov, S. Galeotta, S. Galli, K. Ganga, R. T. Génova-Santos, M. Gerbino, T. Ghosh, J. González-Nuevo, K. M. Górski,S. Gratton, A. Gruppuso, J. E. Gudmundsson, J. Hamann, W. Handley, D. Herranz, E. Hivon, Z. Huang, A. H. Jaffe, W. C. Jones, A. Karakci, E. Keihänen,R. Keskitalo, K. Kiiveri, J. Kim, T. S. Kisner, L. Knox, N. Krachmalnicoff, M. Kunz, H. Kurki-Suonio, G. Lagache, J.- M. Lamarre, A. Lasenby, M. Lattanzi, C. R. Lawrence, M. Le Jeune, P. Lemos, J. Lesgourgues, F. Levrier, A. Lewis, M. Liguori, P. B. Lilje, M. Lilley, V. Lindholm, M. López-Caniego, P. M. Lubin, Y.-Z. Ma, J. F. Mac´ıas-Pérez,G. Maggio, D. Maino, N. Mandolesi, A. Mangilli, A. Marcos- Caballero, M. Maris, P. G. Martin, M. Martinelli, E. Mart´ınez-González, S. Matarrese, N. Mauri, J. D. McEwen, P. R. Meinhold, A. Melchiorri, A. Mennella, M. Migliaccio, M. Millea, S. Mitra, M.-A. Miville-Deschênes, D. Molinari, L. Montier, G. Morgante, A. Moss, P. Natoli, H. U. Nørgaard- Nielsen, L. Pagano, D. Paoletti, B. Partridge, G. Patanchon, H. V. Peiris, F. Perrotta, V. Pettorino, F. Piacentini, L. Polastri, G. Polenta, J.-L. Puget, J. P. Rachen, M. Reinecke, M. Remazeilles, A. Renzi, G. Rocha, C. Rosset, G. Roudier, J. A. Rubiño-Mart´ın,B. Ruiz-Granados, L. Salvati, M. Sandri, M. Savelainen, D. Scott, E. P. S. Shellard, C. Sirignano, G. Sirri, L. D. Spencer, R. Sunyaev, A.-S. Suur-Uski, J. A. Tauber, D. Tavagnacco, BIBLIOGRAPHY 153

M. Tenti, L. Toﬀolatti, M. Tomasi, T. Trombetti, L. Valenziano, J. Valivi- ita, B. Van Tent, L. Vibert, P. Vielva, F. Villa, N. Vittorio, B. D. Wandelt, I. K. Wehus, M. White, S. D. M. White, A. Zacchei, and A. Zonca. Planck 2018 results. VI. Cosmological parameters. ArXiv e-prints, July 2018.

[23] S. Weinberg. Cosmology. Oxford University Press, 2008.

[24] S. Weinberg. The Cosmological Constant Problems (Talk given at Dark Matter 2000, February, 2000). ArXiv Astrophysics e-prints, May 2000.

[25] K. Garrett and G. D¯uda. Dark Matter: A Primer. Advances in Astronomy, 2011:968283, 2011.

[26] Dmitry S. Gorbunov and V. A. Rubakov. Introduction to the theory of the early universe. 2011.

[27] A. Ijjas, P. J. Steinhardt, and A. Loeb. Inﬂationary paradigm in trouble after Planck2013. Physics Letters B, 723:261–266, June 2013.

[28] H. W. Babcock. The rotation of the Andromeda Nebula. Lick Observatory Bulletin, 19:41–51, 1939.

[29] M. S. Roberts and R. N. Whitehurst. The rotation curve and geometry of M31 at large galactocentric distances. The Astrophysical Journal, 201:327– 346, October 1975.

[30] C. Carignan, L. Chemin, W. K. Huchtmeier, and F. J. Lockman. The Extended H I Rotation Curve and Mass Distribution of M31. The Astro- physical Journal Letters, 641:L109–L112, April 2006.

[31] K. C. Freeman. On the Disks of Spiral and S0 Galaxies. The Astrophysical Journal, 160:811, June 1970.

[32] A. Bosma. The distribution and kinematics of neutral hydrogen in spiral 154 BIBLIOGRAPHY

galaxies of various morphological types. PhD thesis, PhD Thesis, Groningen Univ., (1978), 1978.

[33] V. C. Rubin, W. K. Ford, Jr., and N. Thonnard. Extended rotation curves of high-luminosity spiral galaxies. IV - Systematic dynamical properties, SA through SC. The Astrophysical Journal Letters, 225:L107–L111, November 1978.

[34] W. Dehnen and J. Binney. Mass models of the Milky Way. Monthly Notices of the Royal Astronomical Society, 294:429, March 1998.

[35] Y. Sofue, M. Honma, and T. Omodaka. Uniﬁed Rotation Curve of the Galaxy – Decomposition into de Vaucouleurs Bulge, Disk, Dark Halo, and the 9-kpc Rotation Dip –. Publications of the ASJ, 61:227–236, February 2009.

[36] M. Weber and W. de Boer. Determination of the local dark matter density in our Galaxy. Astronomy and Astrophysics, 509:A25, January 2010.

[37] M. Pato, F. Iocco, and G. Bertone. Dynamical constraints on the dark matter distribution in the Milky Way. Journal of Cosmology and Astroparticle Physics, 12:001, December 2015.

[38] G. Bertone and D. Hooper. A History of Dark Matter. ArXiv e-prints, May 2016.

[39] E. Hubble and M. L. Humason. The Velocity-Distance Relation among Extra-Galactic Nebulae. The Astrophysical Journal, 74:43, July 1931.

[40] F. Zwicky. On the Masses of Nebulae and of Clusters of Nebulae. The Astrophysical Journal, 86:217, October 1937. BIBLIOGRAPHY 155

[41] Elisabet Liljeblad. Estimating Masses of Galaxy Clusters. Master’s thesis, Chalmers University of Technology, Gothenburg, Sweden, 2012.

[42] R. Sadat. Clusters of Galaxies and Mass Estimates. In D. Valls-Gabaud, M. A. Hendry, P. Molaro, and K. Chamcham, editors, From Quantum Fluc- tuations to Cosmological Structures, volume 126 of Astronomical Society of the Paciﬁc Conference Series, page 349, September 1997.

[43] R. Massey, T. Kitching, and J. Richard. The dark matter of gravitational lensing. Reports on Progress in Physics, 73(8):086901, August 2010.

[44] E. J. Hallman and M. Markevitch. Chandra Observation of the Merging Cluster A168: A Late Stage in the Evolution of a Cold Front. The Astro- physical Journal Letters, 610:L81–L84, August 2004.

[45] D. Clowe, M. Bradaˇc,A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones, and D. Zaritsky. A Direct Empirical Proof of the Existence of Dark Matter. The Astrophysical Journal Letters, 648:L109–L113, Septem- ber 2006.

[46] D. J. Fixsen, E. S. Cheng, D. A. Cottingham, R. E. Eplee, Jr., R. B. Isaacman, J. C. Mather, S. S. Meyer, P. D. Noerdlinger, R. A. Shafer, R. Weiss, E. L. Wright, C. L. Bennett, N. W. Boggess, T. Kelsall, S. H. Moseley, R. F. Silverberg, G. F. Smoot, and D. T. Wilkinson. Cosmic microwave background dipole spectrum measured by the COBE FIRAS instrument. The Astrophysical Journal, 420:445–449, January 1994.

[47] G. F. Smoot. Cosmic Microwave Background Anisotropy Large Angular Scales / COBE DMR Results. In N. Sanchez and A. Zichichi, editors, NATO Advanced Science Institutes (ASI) Series C, volume 467 of NATO Advanced Science Institutes (ASI) Series C, pages 301–311, 1995. 156 BIBLIOGRAPHY

[48] D. N. Spergel, L. Verde, H. V. Peiris, E. Komatsu, M. R. Nolta, C. L. Bennett, M. Halpern, G. Hinshaw, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer, L. Page, G. S. Tucker, J. L. Weiland, E. Wollack, and E. L. Wright. First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters. Astrophysical Journal Supple- ments, 148:175–194, September 2003.

[49] D. N. Spergel, R. Bean, O. Dor´e,M. R. Nolta, C. L. Bennett, J. Dunk- ley, G. Hinshaw, N. Jarosik, E. Komatsu, L. Page, H. V. Peiris, L. Verde, M. Halpern, R. S. Hill, A. Kogut, M. Limon, S. S. Meyer, N. Odegard, G. S. Tucker, J. L. Weiland, E. Wollack, and E. L. Wright. Three-Year Wilkin- son Microwave Anisotropy Probe (WMAP) Observations: Implications for Cosmology. Astrophysical Journal Supplements, 170:377–408, June 2007.

[50] U. Seljak and M. Zaldarriaga. A Line-of-Sight Integration Approach to Cosmic Microwave Background Anisotropies. The Astrophysical Journal, 469:437, October 1996.

[51] Antony Lewis and Sarah Bridle. Cosmological parameters from CMB and other data: A Monte Carlo approach. Phys. Rev. D, 66:103511, 2002.

[52] J. Lesgourgues. The Cosmic Linear Anisotropy Solving System (CLASS) I: Overview. ArXiv e-prints, April 2011.

[53] Planck Collaboration, Aghanim, N., Arnaud, M., Ashdown, M., Aumont, J., Baccigalupi, C., Banday, A. J., Barreiro, R. B., Bartlett, J. G., Bartolo, N., Battaner, E., Benabed, K., Benoˆıt,A., Benoit-L´evy, A., Bernard, J.-P., Bersanelli, M., Bielewicz, P., Bock, J. J., Bonaldi, A., Bonavera, L., Bond, J. R., Borrill, J., Bouchet, F. R., Boulanger, F., Bucher, M., Burigana, C., Butler, R. C., Calabrese, E., Cardoso, J.-F., Catalano, A., Challinor, A., BIBLIOGRAPHY 157

Chiang, H. C., Christensen, P. R., Clements, D. L., Colombo, L. P. L., Combet, C., Coulais, A., Crill, B. P., Curto, A., Cuttaia, F., Danese, L., Davies, R. D., Davis, R. J., de Bernardis, P., de Rosa, A., de Zotti, G., De- labrouille, J., Désert,F.-X., Di Valentino, E., Dickinson, C., Diego, J. M., Dolag, K., Dole, H., Donzelli, S., Doré,O., Douspis, M., Ducout, A., Dunk- ley, J., Dupac, X., Efstathiou, G., Elsner, F., Enßlin, T. A., Eriksen, H. K., Fergusson, J., Finelli, F., Forni, O., Frailis, M., Fraisse, A. A., Franceschi, E., Frejsel, A., Galeotta, S., Galli, S., Ganga, K., Gauthier, C., Gerbino, M., Giard, M., Gjerløw, E., González-Nuevo, J., Górski,K. M., Gratton, S., Gregorio, A., Gruppuso, A., Gudmundsson, J. E., Hamann, J., Hansen, F. K., Harrison, D. L., Helou, G., Henrot-Versillé,S., Hernández-Monteagudo, C., Herranz, D., Hildebrandt, S. R., Hivon, E., Holmes, W. A., Hornstrup, A., Huffenberger, K. M., Hurier, G., Jaffe, A. H., Jones, W. C., Juvela, M., Keihänen,E., Keskitalo, R., Kiiveri, K., Knoche, J., Knox, L., Kunz, M., Kurki-Suonio, H., Lagache, G., Lähteenmäki,A., Lamarre, J.-M., Lasenby, A., Lattanzi, M., Lawrence, C. R., Le Jeune, M., Leonardi, R., Lesgour- gues, J., Levrier, F., Lewis, A., Liguori, M., Lilje, P. B., Lilley, M., Linden- Vørnle, M., Lindholm, V., López-Caniego, M., Mac´ıas-Pérez,J. F., Maffei, B., Maggio, G., Maino, D., Mandolesi, N., Mangilli, A., Maris, M., Martin, P. G., Mart´ınez-González,E., Masi, S., Matarrese, S., Meinhold, P. R., Mel- chiorri, A., Migliaccio, M., Millea, M., Mitra, S., Miville-Deschênes,M.-A., Moneti, A., Montier, L., Morgante, G., Mortlock, D., Mottet, S., Munshi, D., Murphy, J. A., Narimani, A., Naselsky, P., Nati, F., Natoli, P., Noviello, F., Novikov, D., Novikov, I., Oxborrow, C. A., Paci, F., Pagano, L., Pajot, F., Paoletti, D., Partridge, B., Pasian, F., Patanchon, G., Pearson, T. J., Perdereau, O., Perotto, L., Pettorino, V., Piacentini, F., Piat, M., Pier- paoli, E., Pietrobon, D., Plaszczynski, S., Pointecouteau, E., Polenta, G., 158 BIBLIOGRAPHY

Ponthieu, N., Pratt, G. W., Prunet, S., Puget, J.-L., Rachen, J. P., Rei- necke, M., Remazeilles, M., Renault, C., Renzi, A., Ristorcelli, I., Rocha, G., Rossetti, M., Roudier, G., RouillédÓrfeuil, B., Rubiño-Mart´ın,J. A., Rusholme, B., Salvati, L., Sandri, M., Santos, D., Savelainen, M., Savini, G., Scott, D., Serra, P., Spencer, L. D., Spinelli, M., Stolyarov, V., Stom- por, R., Sunyaev, R., Sutton, D., Suur-Uski, A.-S., Sygnet, J.-F., Tauber, J. A., Terenzi, L., Toffolatti, L., Tomasi, M., Tristram, M., Trombetti, T., Tucci, M., Tuovinen, J., Umana, G., Valenziano, L., Valiviita, J., Van Tent, F., Vielva, P., Villa, F., Wade, L. A., Wandelt, B. D., Wehus, I. K., Yvon, D., Zacchei, A., and Zonca, A. Planck 2015 results - xi. cmb power spectra, likelihoods, and robustness of parameters. A&A, 594:A11, 2016.

[54] J. L. Feng. Dark Matter Candidates from Particle Physics and Methods of Detection. Annual Review of Astronomy and Astrophysics, 48:495–545, September 2010.

[55] R. N. Mohapatra. Supersymmetry and R-parity: an overview. Physica Scripta, 90(8):088004, August 2015.

[56] L. Bergstr¨om. Dark matter candidates. New Journal of Physics, 11(10):105006, October 2009.

[57] Xiangyi Cui, Abdusalam Abdukerim, Wei Chen, Xun Chen, Yunhua Chen, Binbin Dong, Deqing Fang, Changbo Fu, Karl Giboni, Franco Giuliani, Linhui Gu, Yikun Gu, Xuyuan Guo, Zhifan Guo, Ke Han, Changda He, Di Huang, Shengming He, Xingtao Huang, Zhou Huang, Xiangdong Ji, Yonglin Ju, Shaoli Li, Yao Li, Heng Lin, Huaxuan Liu, Jianglai Liu, Yugang Ma, Yajun Mao, Kaixiang Ni, Jinhua Ning, Xiangxiang Ren, Fang Shi, Andi Tan, Cheng Wang, Hongwei Wang, Meng Wang, Qiuhong Wang, Siguang BIBLIOGRAPHY 159

Wang, Xiuli Wang, Xuming Wang, Qinyu Wu, Shiyong Wu, Mengjiao Xiao, Pengwei Xie, Binbin Yan, Yong Yang, Jianfeng Yue, Dan Zhang, Hong- guang Zhang, Tao Zhang, Tianqi Zhang, Li Zhao, Jifang Zhou, Ning Zhou, Xiaopeng Zhou, and PandaX-II Collaboration.

[58] C. Renault, C. Afonso, E. Aubourg, P. Bareyre, F. Bauer, S. Brehin, C. Coutures, C. Gaucherel, J. F. Glicenstein, B. Goldman, M. Gros, D. Hardin, J. de Kat, M. Lachieze-Rey, B. Laurent, E. Lesquoy, C. Mag- neville, A. Milsztajn, L. Moscoso, N. Palanque-Delabrouille, F. Queinnec, J. Rich, M. Spiro, L. Vigroux, S. Zylberajch, R. Ansari, F. Cavalier, F. Cou- chot, B. Mansoux, M. Moniez, O. Perdereau, J. P. Beaulieu, R. Ferlet, P. Grison, A. Vidal-Madjar, J. Guibert, O. Moreau, E. Maurice, L. Pre- vot, C. Gry, S. Char, and J. Fernandez. Observational limits on MACHOS in the Galactic Halo. Astronomy and Astrophysics, 324:L69–L72, August 1997.

[59] P. Tisserand, L. Le Guillou, C. Afonso, J. N. Albert, J. Andersen, R. Ansari, E.´ Aubourg, P. Bareyre, J. P. Beaulieu, X. Charlot, C. Coutures, R. Ferlet, P. Fouqu´e,J. F. Glicenstein, B. Goldman, A. Gould, D. Graﬀ, M. Gros, J. Haissinski, C. Hamadache, J. de Kat, T. Lasserre, E.´ Lesquoy, C. Loup, C. Magneville, J. B. Marquette, E.´ Maurice, A. Maury, A. Milsztajn, M. Moniez, N. Palanque-Delabrouille, O. Perdereau, Y. R. Rahal, J. Rich, M. Spiro, A. Vidal- Madjar, L. Vigroux, S. Zylberajch, and EROS-2 Col- laboration. Limits on the Macho content of the Galactic Halo from the EROS-2 Survey of the Magellanic Clouds. Astronomy and Astrophysics, 469:387–404, July 2007.

[60] Yacine Ali-Ha¨ımoud and Marc Kamionkowski. Cosmic microwave background limits on accreting primordial black holes. Phys. Rev. D, 95:043534, 160 BIBLIOGRAPHY

Feb 2017.

[61] Sai Wang, Yi-Fan Wang, Qing-Guo Huang, and Tjonnie G. F. Li. Con- straints on the primordial black hole abundance from the ﬁrst advanced ligo observation run using the stochastic gravitational-wave background. Phys. Rev. Lett., 120:191102, May 2018.

[62] B. Carr, F. K¨uhnel,and M. Sandstad. Primordial black holes as dark matter. Phys. Rev. D, 94(8):083504, October 2016.

[63] S. Calchi Novati, S. Mirzoyan, Ph. Jetzer, and G. Scarpetta. Microlensing towards the smc: a new analysis of ogle and eros results. Monthly Notices of the Royal Astronomical Society, 435(2):1582–1597, 2013.

[64] E. Mediavilla, J. A. Mu˜noz,E. Falco, V. Motta, E. Guerras, H. Canovas, C. Jean, A. Oscoz, and A. M. Mosquera. Microlensing-based estimate of the mass fraction in compact objects in lens galaxies. The Astrophysical Journal, 706(2):1451, 2009.

[65] Savvas M. Koushiappas and Abraham Loeb. Dynamics of dwarf galaxies disfavor stellar-mass black holes as dark matter. Phys. Rev. Lett., 119:041102, Jul 2017.

[66] R. D. Peccei and H. R. Quinn. Constraints imposed by CP conservation in the presence of pseudoparticles. Phys. Rev. D, 16:1791–1797, September 1977.

[67] S. Weinberg. The U(1) problem. Phys. Rev. D, 11:3583–3593, June 1975.

[68] G. ’t Hooft. Symmetry Breaking through Bell-Jackiw Anomalies. Physical Review Letters, 37:8–11, July 1976.

[69] E. W. Kolb and M. S. Turner. The early universe. 1990. BIBLIOGRAPHY 161

[70] J. M. Pendlebury, S. Afach, N. J. Ayres, C. A. Baker, G. Ban, G. Bi- son, K. Bodek, M. Burghoff, P. Geltenbort, K. Green, W. C. Griffith, M. van der Grinten, Z. D. Grujić,P. G. Harris, V. Hélaine,P. Iaydjiev, S. N. Ivanov, M. Kasprzak, Y. Kermaidic, K. Kirch, H.-C. Koch, S. Kom- posch, A. Kozela, J. Krempel, B. Lauss, T. Lefort, Y. Lemière, D. J. R. May, M. Musgrave, O. Naviliat-Cuncic, F. M. Piegsa, G. Pignol, P. N. Prashanth, G. Quéméner,M. Rawlik, D. Rebreyend, J. D. Richardson, D. Ries, S. Roccia, D. Rozpedzik, A. Schnabel, P. Schmidt-Wellenburg, N. Severijns, D. Shiers, J. A. Thorne, A. Weis, O. J. Winston, E. Wursten, J. Zejma, and G. Zsigmond. Revised experimental upper limit on the electric dipole moment of the neutron. Phys. Rev. D, 92(9):092003, November 2015.

[71] R. D. Peccei and H. R. Quinn. CP conservation in the presence of pseudoparticles. Physical Review Letters, 38:1440–1443, June 1977.

[72] S. Weinberg. A new light boson? Physical Review Letters, 40:223–226, January 1978.

[73] F. Wilczek. Problem of strong P and T invariance in the presence of in- stantons. Physical Review Letters, 40:279–282, January 1978.

[74] P. Sikivie. Experimental tests of the ’invisible’ axion. Physical Review Letters, 51:1415–1417, October 1983.

[75] J. E. Kim. Weak-interaction singlet and strong CP invariance. Physical Review Letters, 43:103–107, July 1979.

[76] M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov. Can conﬁnement ensure natural CP invariance of strong interactions? Nuclear Physics B, 166:493–506, April 1980. 162 BIBLIOGRAPHY

[77] M. Dine, W. Fischler, and M. Srednicki. A simple solution to the strong CP problem with a harmless axion. Physics Letters B, 104:199–202, August 1981.

[78] A. R. Zhitnitsky. On Possible Suppression of the Axion Hadron Interactions. (In Russian). Sov. J. Nucl. Phys., 31:260, 1980. [Yad. Fiz.31,497(1980)].

[79] P. Sikivie. Axion Cosmology. In M. Kuster, G. Raﬀelt, and B. Beltr´an, editors, Axions, volume 741 of Lecture Notes in Physics, Berlin Springer Verlag, page 19, 2008.

[80] Y. Semertzidis and J. D. Vergados. Modulation and diurnal variation in axionic dark matter searches. ArXiv e-prints, November 2015.

[81] O. Wantz and E. P. S. Shellard. Axion cosmology revisited. Phys. Rev. D, 82(12):123508, December 2010.

[82] R. A. Battye and E. P. S. Shellard. Axion string constraints. Physical Review Letters, 73:2954–2957, November 1994.

[83] P. W. Graham, I. G. Irastorza, S. K. Lamoreaux, A. Lindner, and K. A. van Bibber. Experimental Searches for the Axion and Axion-Like Particles. Annual Review of Nuclear and Particle Science, 65:485–514, October 2015.

[84] CERN. Cast experiment. http://cast.web.cern.ch/CAST/CAST.php, 2015.

[85] V. Anastassopoulos, S. Aune, Barth, et al. New CAST limit on the axion- photon interaction. Nature Physics, 13:584–590, June 2017.

[86] University of Washington. Admx experiment. http://depts.washington. edu/admx/about-us.shtml, 2015. BIBLIOGRAPHY 163

[87] University of Washington. Admx experiment. http://depts.washington. edu/admx/receiver.shtml, 2015.

[88] University of Washington. Admx experiment. http://depts.washington. edu/admx/cavity.shtml, 2015.

[89] S. J. Asztalos, G. Carosi, C. Hagmann, D. Kinion, K. van Bibber, M. Hotz, L. J. Rosenberg, G. Rybka, J. Hoskins, J. Hwang, P. Sikivie, D. B. Tan- ner, R. Bradley, J. Clarke, and ADMX Collaboration. SQUID-Based Mi- crowave Cavity Search for Dark-Matter Axions. Physical Review Letters, 104(4):041301, January 2010.

[90] Ben Brubaker-Yale University. Admx hf. https://conferences.pa.ucla. edu/dm16/talks/brubaker.pdf, 2016.

[91] A. Garcon, D. Aybas, J. W. Blanchard, G. Centers, N. L. Figueroa, P. W. Graham, D. F. J. Kimball, S. Rajendran, M. Gil Sendra, A. O. Sushkov, L. Trahms, T. Wang, A. Wickenbrock, T. Wu, and D. Budker. The Cosmic Axion Spin Precession Experiment (CASPEr): a dark-matter search with nuclear magnetic resonance. ArXiv e-prints, July 2017.

[92] M. Betz, F. Caspers, M. Gasior, M. Thumm, and S. W. Rieger. First results of the CERN Resonant Weakly Interacting sub-eV Particle Search (CROWS). Phys. Rev. D, 88(7):075014, October 2013.

[93] W. Heisenberg and H. Euler. Consequences of Dirac Theory of the Positron. ArXiv Physics e-prints, May 2006.

[94] L. Maiani, R. Petronzio, and E. Zavattini. Eﬀects of nearly massless, spin- zero particles on light propagation in a magnetic ﬁeld. Physics Letters B, 175:359–363, August 1986. 164 BIBLIOGRAPHY

[95] A. Ringwald. Axion interpretation of the PVLAS data? In A. Bottino, E. Coccia, J. Morales, and J. Puimed´onv, editors, Journal of Physics Con- ference Series, volume 39 of Journal of Physics Conference Series, pages 197–199, May 2006.

[96] F. Della Valle, A. Ejlli, U. Gastaldi, G. Messineo, E. Milotti, R. Pengo, G. Ruoso, and G. Zavattini. The PVLAS experiment: measuring vacuum magnetic birefringence and dichroism with a birefringent Fabry-Perot cavity. European Physical Journal C, 76:24, January 2016.

[97] Any Light Particle Search (ALPS). Representation of photon regeneration.

https://alps.desy.de/, 2010.

[98] K. Ehret, M. Frede, Ghazaryan, et al. New ALPS results on hidden-sector lightweights. Physics Letters B, 689:149–155, May 2010.

[99] R. Ballou, G. Deferne, Duvillaret, et al. Latest Results of the OSQAR Photon Regeneration Experiment for Axion-Like Particle Search. ArXiv e-prints, October 2014.

[100] Andreas Ringwald. Limits on alp parameter space. http://inspirehep. net/record/848652/plots, 2010.

[101] A. Ringwald. Challenges and Opportunities for the Next Generation of Photon Regeneration Experiments. In D. B. Tanner and K. A. van Bibber, editors, American Institute of Physics Conference Series, volume 1274 of American Institute of Physics Conference Series, pages 169–174, August 2010.

[102] G. G. Raﬀelt and D. S. P. Dearborn. Bounds on hadronic axions from stellar evolution. Phys. Rev. D, 36:2211–2225, October 1987. BIBLIOGRAPHY 165

[103] G. G. Raffelt. Astrophysical Axion Bounds. In M. Kuster, G. Raffelt, and B. Beltrán,editors, Axions, volume 741 of Lecture Notes in Physics, Berlin Springer Verlag, page 51, 2008.

[104] H. Schlattl, A. Weiss, and G. Raﬀelt. Helioseismological constraint on solar axion emission. Astroparticle Physics, 10:353–359, May 1999.

[105] A. Ayala, I. Dom´ınguez,M. Giannotti, A. Mirizzi, and O. Straniero. Revisit- ing the Bound on Axion-Photon Coupling from Globular Clusters. Physical Review Letters, 113(19):191302, November 2014.

[106] G. G. Raﬀelt. Stars as laboratories for fundamental physics : the astrophysics of neutrinos, axions, and other weakly interacting particles. 1996.

[107] K. Hirata, T. Kajita, M. Koshiba, M. Nakahata, and Y. Oyama. Obser- vation of a neutrino burst from the supernova SN1987A. Physical Review Letters, 58:1490–1493, April 1987.

[108] R. M. Bionta, G. Blewitt, C. B. Bratton, D. Casper, and A. Ciocio. Obser- vation of a neutrino burst in coincidence with supernova 1987A in the Large Magellanic Cloud. Physical Review Letters, 58:1494–1496, April 1987.

[109] R. A. Battye and E. P. S. Shellard. Axion string cosmology and its contro- versies. ArXiv Astrophysics e-prints, September 1999.

[110] D. Cadamuro. Cosmological limits on axions and axion-like particles. PhD thesis, Ludwig Maximilians University - Munich, 2012.

[111] E. Giusarma, E. Di Valentino, M. Lattanzi, A. Melchiorri, and O. Mena. Relic neutrinos, thermal axions, and cosmology in early 2014. Phys. Rev. D, 90(4):043507, August 2014.

[112] BICEP2 Collaboration, P. A. R. Ade, Aikin, et al. Detection of B-Mode 166 BIBLIOGRAPHY

Polarization at Degree Angular Scales by BICEP2. Physical Review Letters, 112(24):241101, June 2014.

[113] E. Di Valentino, E. Giusarma, M. Lattanzi, A. Melchiorri, and O. Mena. Axion cold dark matter: Status after Planck and BICEP2. Phys. Rev. D, 90(4):043534, August 2014.

[114] E. Di Valentino, E. Giusarma, M. Lattanzi, O. Mena, A. Melchiorri, and J. Silk. Cosmological axion and neutrino mass constraints from Planck 2015 temperature and polarization data. Physics Letters B, 752:182–185, January 2016.

[115] A. Melchiorri. A cosmological bound on the thermal axion mass. Nuclear Physics B Proceedings Supplements, 194:100–104, October 2009.

[116] Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Arnaud, F. Arroja, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, and et al. Planck 2015 results. XIX. Constraints on primordial magnetic ﬁelds. Astronomy and Astrophysics, 594:A19, September 2016.

[117] Axel Brandenburg and Kandaswamy Subramanian. Astrophysical magnetic ﬁelds and nonlinear dynamo theory. Physics Reports, 417:1–209, October 2005.

[118] F. Govoni and L. Feretti. Magnetic Fields in Clusters of Galaxies. Inter- national Journal of Modern Physics D, 13:1549–1594, 2004.

[119] Rainer Beck. Galactic and Extragalactic Magnetic Fields. Space Science Reviews, 99:243–260, October 2001.

[120] K. Kelley and P. J. Quinn. A Radio Astronomy Search for Cold Dark Matter Axions. The Astrophysical Journal Letters, 845:L4, August 2017. BIBLIOGRAPHY 167

[121] G. Sigl. Astrophysical Haloscopes. ArXiv e-prints, August 2017.

[122] R. A. Battye, I. W. A. Browne, C. Dickinson, G. Heron, B. Maﬀei, and A. Pourtsidou. H I intensity mapping: a single dish approach. Monthly Notices of the Royal Astronomical Society, 434:1239–1256, September 2013.

[123] A. Bonafede, L. Feretti, M. Murgia, F. Govoni, G. Giovannini, D. Dallacasa, K. Dolag, and G. B. Taylor. The Coma cluster magnetic ﬁeld from Faraday rotation measures. Astronomy and Astrophysics, 513:A30, April 2010.

[124] D. Guidetti, M. Murgia, F. Govoni, P. Parma, L. Gregorini, H. R. de Ruiter, R. A. Cameron, and R. Fanti. The intracluster magnetic ﬁeld power spectrum in Abell 2382. Astronomy and Astrophysics, 483:699–713, June 2008.

[125] G. Rave, D. Kushnir, and E. Waxman. What can we really learn about Magnetic Fields in Galaxy Clusters from Faraday Rotation observations? ArXiv e-prints, April 2013.

[126] A. Caputo, C. Pe˜naGaray, and S. J. Witte. Looking for Axion Dark Matter in Dwarf Spheroidals. ArXiv e-prints, May 2018.

[127] K. Kelley and P. J. Quinn. Searching for Axion Dark Matter using Radio Telescopes. ArXiv e-prints, October 2017.

[128] B. D. Blout, E. J. Daw, M. P. Decowski, P. T. P. Ho, L. J. Rosenberg, and D. B. Yu. A Radio Telescope Search for Axions. The Astrophysical Journal, 546:825–828, January 2001.

[129] M. L. Mateo. Dwarf Galaxies of the Local Group. Annual Review of As- tronomy and Astrophysics, 36:435–506, 1998.

[130] S. Chandrasekhar. Principles of stellar dynamics. 1960.

[131] Alma X. Gonz´alez-Morales,David J. E. Marsh, Jorge Pe˜narrubia,and 168 BIBLIOGRAPHY

Luis A. Ure˜na-L´opez. Unbiased constraints on ultralight axion mass from dwarf spheroidal galaxies. Monthly Notices of the Royal Astronomical So- ciety, 472(2):1346–1360, 2017.

[132] CristóbalSifón,Felipe Menanteau, Matthew Hasselfield, Tobias A. Mar- riage, John P. Hughes, L. Felipe Barrientos, Jorge González, Leopoldo In- fante, Graeme E. Addison, Andrew J. Baker, Nick Battaglia, J. Richard Bond, Devin Crichton, Sudeep Das, Mark J. Devlin, Joanna Dunk- ley, Rolando Dünner, Megan B. Gralla, Amir Hajian, Matt Hilton, Adam D. Hincks, Arthur B. Kosowsky, Danica Marsden, Kavilan Moodley, Michael D. Niemack, Michael R. Nolta, Lyman A. Page, Bruce Partridge, Erik D. Reese, Neelima Sehgal, Jon Sievers, David N. Spergel, Suzanne T. Staggs, Robert J. Thornton, Hy Trac, and Edward J. Wollack. The Ata- cama Cosmology Telescope: Dynamical Masses and Scaling Relations for a Sample of Massive Sunyaev-Zel’dovich Effect Selected Galaxy Clusters. The Astrophysical Journal, 772:25, July 2013.

[133] Jounghun Lee, Suk Kim, and Soo-Chang Rey. A New Dynamical Mass Measurement for the VIRGO Cluster Using the Radial Velocity Proﬁle of the Filament Galaxies. The Astrophysical Journal, 807:122, July 2015.

[134] P. Fouqu´e,J. M. Solanes, T. Sanchis, and C. Balkowski. Structure, mass and distance of the Virgo cluster from a Tolman-Bondi model. Astronomy and Astrophysics, 375:770–780, September 2001.

[135] Jeﬀrey M. Kubo, Albert Stebbins, James Annis, Ian P. Dell’Antonio, Huan Lin, Hossein Khiabanian, and Joshua A. Frieman. The Mass of the Coma Cluster from Weak Lensing in the Sloan Digital Sky Survey. The Astro- physical Journal, 671:1466–1470, December 2007. BIBLIOGRAPHY 169

[136] Bjarne Thomsen, William A. Baum, Mark Hammergren, and Guy Worthey. The Distance to the Coma Cluster from Surface Brightness Fluctuations. The Astrophysical Journal, 483:L37–L40, July 1997.

[137] J. F. Navarro, C. S. Frenk, and S. D. M. White. The Structure of Cold Dark Matter Halos. The Astrophysical Journal, 462:563, May 1996.

[138] A. D. Ludlow, J. F. Navarro, R. E. Angulo, M. Boylan-Kolchin, V. Springel, C. Frenk, and S. D. M. White. The mass-concentration-redshift relation of cold dark matter haloes. Monthly Notices of the Royal Astronomical Society, 441:378–388, June 2014.

[139] D. R. Lorimer and M. Kramer. Handbook of Pulsar Astronomy. October 2012.

[140] G. Raﬀelt and L. Stodolsky. Mixing of the photon with low-mass particles. Phys. Rev. D, 37:1237–1249, March 1988.

[141] A. Hook, Y. Kahn, B. R. Safdi, and Z. Sun. Radio Signals from Axion Dark Matter Conversion in Neutron Star Magnetospheres. ArXiv e-prints, April 2018.

[142] F. P. Huang, K. Kadota, T. Sekiguchi, and H. Tashiro. The radio telescope search for the resonant conversion of cold dark matter axions from the magnetized astrophysical sources. ArXiv e-prints, March 2018.

[143] M. S. Pshirkov and S. B. Popov. Conversion of dark matter axions to photons in magnetospheres of neutron stars. Soviet Journal of Experimental and Theoretical Physics, 108:384–388, March 2009.

[144] V. S. Beskin, Aleksandr V. Gurevich, and Ya. N. Istomin. Physics of the pulsar magnetosphere. 1993. 170 BIBLIOGRAPHY

[145] D. Lai and J. Heyl. Probing axions with radiation from magnetic stars. Phys. Rev. D, 74(12):123003, December 2006.

[146] P. Goldreich and W. H. Julian. Pulsar Electrodynamics. The Astrophysical Journal, 157:869, August 1969.

[147] V. M. Vasyliunas. Relation between magnetic ﬁelds and electric currents in plasmas. Annales Geophysicae, 23(7):2589–2597, 2005.

[148] D. B. Melrose and R. Yuen. Pulsar electrodynamics: an unsolved problem. Journal of Plasma Physics, 82(2):635820202, 2016.

[149] Curt Wittig. The landau- zener formula. The Journal of Physical Chemistry B, 109(17):8428–8430, 2005.

[150] M. E. Tobar, B. T. McAllister, and M. Goryachev. Modiﬁed Axion Elec- trodynamics through Oscillating Vacuum Polarization and Magnetization and Low Mass Detection. ArXiv e-prints, September 2018.