Master of Science Thesis
Thermalisation of inelastic dark matter in the Sun with a light mediator
Simon Israelsson
Particle and Astroparticle Physics, Department of Physics, School of Engineering Sciences, KTH Royal Institute of Technology, SE-106 91 Stockholm, Sweden
Stockholm, Sweden 2018 Typeset in LATEX
Examensarbetesuppsats f¨or avl¨aggande av Masterexamen i Teknisk fysik, med in- riktning mot Teoretisk fysik. Master’s thesis for a Master’s degree in Engineering Physics in the subject area of Theoretical physics.
TRITA-SCI-GRU 2018:308
c Simon Israelsson, August 2018 Printed in Sweden by Universitetsservice US AB Abstract
Particle dark matter is a popular solution to the missing mass problem present in the Universe. If dark matter interacts with ordinary matter, even very weakly, it might be the case that it is captured and accumulated in the Sun, where it may then annihilate into particles that we can observe here on Earth. The interaction between dark matter and standard model particles may be mediated by a light dark sector particle. This would introduce an extra recoil energy suppression into the scattering cross section for collision events, which is of the form needed to possibly also alleviate some of the observed small scale structure issues of collisionless cold dark matter. In this work we perform numerical simulations of the capture and subsequent scattering of inelastic dark matter in the Sun, in the presence of a light mediator particle. We find that the presence of the mediator results in a narrower capture region than expected without it and that it mainly affects the scattering rate in the phase space region where the highest scattering rates are found. How- ever, it is not seen to cause any noticeable difference to the radial distribution of dark matter in the Sun. No steady state is reached and the captured dark matter does not reach thermal equilibrium in the Sun.
Key words: dark matter, solar capture, inelastic dark matter, light mediator.
iii Sammanfattning
M¨orka materiapartiklar ¨ar en popul¨ar l¨osning till avsaknaden av materia i uni- versum. Om m¨ork materia interagerar med vanlig materia, ¨aven v¨aldigt svagt, s˚askulle det kunna vara fallet att de f˚angasin och ansamlas i solen d¨ar de se- dan f¨orintas och resulterar i partiklar vi kan observera h¨ar p˚ajorden. Interaktio- nen mellan m¨ork materia och vanlig materia skulle kunna f¨ormedlas av en l¨att kraftb¨arare i den m¨orka sektorn. Detta skulle introducera ett rekylenergi-beroende i spridningstv¨arsnittet f¨or kollisioner av samma form som beh¨ovs f¨or att potentiellt avhj¨alpa ett antal observerade problem som i dagsl¨aget finns f¨or kollisionsl¨os kall m¨ork materia. I detta arbete har vi utf¨ort numeriska simuleringar av inf˚angningen och den efterf¨oljande spridningen av m¨ork materia inuti solen d˚aen l¨att kraftb¨arare ¨ar n¨arvarande. Vi kommer fram till att den l¨atta kraftb¨araren resulterar i ett mindre inf˚angningsomr˚ade ¨an vad man v¨antar sig utan dess existens, samt att den p˚averkar spridningen som mest i omr˚adenav fasrummet d¨ar den totala spridningen ¨ar som st¨orst. D¨aremot s˚averkar den inte leda till n˚agonurskiljbar skillnad f¨or den radi- ella distributionen av m¨ork materia i solen. Inget station¨art beteende n˚asoch den inf˚angadem¨orka materian uppn˚arinte termisk j¨amvikt i solen.
Nyckelord: m¨ork materia, solinf˚angning,inelastisk m¨ork materia, l¨att kraftb¨arare.
iv Preface
Acknowledgements
I would like to begin by thanking my supervisor Tommy Ohlsson for enabling me to work on this project, and Mattias Blennow for making me aware that the project was available. I would also like to thank Stefan Clementz for allowing me to pick up where he left off, for allowing me to inherit his simulation code, and for all the help and guidance he has provided throughout the project. Further I would like to thank my other office mates, Marcus and Anton, who together with Stefan provided an amusing working environment filled with interesting discussions. For their insights provided during the regularly held journal clubs and other discussions, I would also like to thank Florian and Sofiane, as well as Sandhya and Sushant. Lastly I would like to thank my family and friends who have supported me throughout the semester and the entirety of my time at KTH.
v
Contents
Abstract ...... iii Sammanfattning ...... iv
Preface v Acknowledgements ...... v
Contents vii
1 Introduction 1 1.1 Outline ...... 2
2 Background 3 2.1 Dark matter ...... 3 2.2 Observational evidence ...... 3 2.2.1 Rotation curves ...... 4 2.2.2 Weak lensing observations ...... 4 2.2.3 The cosmic microwave background ...... 5 2.3 Alternatives to dark matter ...... 6 2.4 Structure issues of dark matter ...... 6 2.4.1 Missing satellites ...... 6 2.4.2 Cusp versus core ...... 7 2.4.3 Too big to fail ...... 7
3 General theory 9 3.1 Inelastic dark matter ...... 9 3.2 Kinematics ...... 10 3.3 Orbital mechanics ...... 12 3.4 Differential scattering cross section ...... 15
3.5 Decay of the χ2 state ...... 18
vii viii Contents
4 Dark matter in the Sun 21 4.1 Solar capture ...... 22 4.2 Scattering in the Sun ...... 24 4.3 Dark matter distribution in the Sun ...... 24
5 Numerical simulations and results 27 5.1 Numerical setup ...... 27 5.2 Solar capture rate ...... 30 5.3 Solar scattering rate ...... 34 5.4 Time evolved distributions ...... 36
6 Summary and conclusions 45
Bibliography 47 Chapter 1
Introduction
The first half of the 20th century resulted in some of the most successful descrip- tions of our universe as of yet in physics. The development of quantum mechanics, and later quantum field theory, provided excellent descriptions of physics on small scales explaining phenomena involving elementary particles, atoms, and molecules. Meanwhile, Einstein’s theory of general relativity (GR) allowed for us to make ac- curate predictions concerning gravitational effects. It was through gravitational interactions, albeit still working using Newtonian gravity, that scientists started noticing discrepancies in the behaviour of astronomical objects [1]. The conclusion was reached that additional matter had to be unaccounted for in order to describe the observed phenomena. The nature of this matter, other than the fact that it was not visible, was, and to a large extent still is, unknown. The name dark mat- ter (DM), or “dunkle Materie” in German, was given to this missing matter [2]. Discrepancies between theory and observations were not limited to the astrophysi- cal sector. Within the particle physics community clear signs of physics beyond the standard model (SM) had also appeared. For example neutrinos, extremely light and very weakly interacting particles, are predicted by the SM to be massless [3]. This has since been proven to be incorrect as the phenomenon of neutrino oscilla- tions, which was experimentally verified in the late 90s [4] and later confirmed in the early 21st century [5], requires there to be a mass difference between the three neutrino mass eigenstates [6]. Astrophysics and particle physics now often go hand in hand attempting to resolve both issues in one go. Particle physicists attempt to extend the SM and resolve its issues while also incorporating candidates for DM and the dark sector in general, and the astrophysicists can deduce, as well as measure, astrophysical observables that constrain the particle physics models. Attempts to further probe this new dark sector have yielded no positive results and the nature of DM, beyond its gravitational interactions, is still unknown. Indirect searches for DM aim to explore this new sector by observing particle and radiation by-products due to DM interactions. A target for these types of observations is the Sun. Mas- sive bodies, such as the Sun, that generate significant gravitational potentials may
1 2 Chapter 1. Introduction capture DM particles on the occasion that they lose sufficient energy in a scattering event while passing through the body [7]. A class of DM known as inelastic DM was originally introduced to alleviate tension between the DAMA experiment [8] and the CDMS experiment [9] as they had obtained disagreeing results. Inelastic DM consists of at least two different DM states that are separated by a very small mass difference that may have significant effects on the scattering kinematics of the DM particles [10]. This thesis is concerned with the thermalisation of inelastic DM in the Sun, where the heavier state is unstable and subsequently decays. A light mediator particle is also included to mediate the scattering process between DM and target particles. For the case where both states are stable and the scattering events are point like it has previously been shown [11] that neither a steady state or equilibrium is reached in general.
1.1 Outline
This thesis is organized as follows: In chapter 2 we cover some general background material concerning DM. In chapter 3 we review the basics of the DM framework known as inelastic DM and we cover most of the necessary theory for this thesis. Moving on to chapter 4 we concern ourselves with the connection between DM and the Sun. Finally, we present the results of this project in chapter 5 and summarise and conclude the thesis in chapter 6. Chapter 2
Background
2.1 Dark matter
Even though the evidence for the existence of DM, covered briefly below, is rather convincing, the true nature of what exactly it is remains unknown. The list of candidates that have been proposed is very long and include, amongst many other, SM neutrinos, particles from supersymmetry, and axions. Popular candidates have historically been weakly interacting massive particles, usually referred to as WIMPs. These DM candidates would be produced in the hot early Universe and “freeze out” such that their abundance today is rather predictable. It turns out that if one assumes DM to have a weak scale interaction cross section as well as just above weak scale mass, the predicted abundance match observed values rather nicely, something that have been dubbed the “WIMP miracle” [12, 13]. Originally WIMPs were those candidates that interacted with the SM through the weak interaction, but the term is now used in a wider sense. Based on cosmological simulations of structure formation in the Universe, DM, whatever it is, is “cold”, i.e., it was non-relativistic after “freeze out”. This in contrast to “hot” DM which would be relativistic instead [1]. Regardless of where the search end up, determining the details and specifics of DM is one of the remaining key discoveries to better understanding the Universe around us.
2.2 Observational evidence
Quite some time has passed since scientist first noticed that something was off with the matter content of the Universe. During the late 19th and early 20th century, galaxy clusters were observed and their masses were estimated [1]. This could be done in several ways. One could simply try to estimate the number of galaxies present in a cluster and multiply it by some estimated galaxy mass, or one could try to apply the virial theorem to the clusters, relating the velocity dispersion of the
3 4 Chapter 2. Background galaxies with the total mass of the cluster. This was famously done by Fritz Zwicky who compared these methods and found that the expected velocity dispersion was an order of magnitude smaller than what was observed. He concluded that an unseen additional mass had to be missing somewhere [2].
2.2.1 Rotation curves Galactic rotation curves, i.e., plots of a galaxy’s circular velocity as a function of radial distance from the galactic centre, provided some of the first convincing evidence for DM [1, 13, 14]. Following classical Newtonian gravity we have the following relation for a spherically symmetric mass distribution r GM(r) v (r) = , (2.1) c r where vc is the circular velocity of the galaxy, G the gravitational constant, and M(r) the enclosed mass within a radius r. If we were to move far out enough that we include all the visible stars and gas in the galaxy we would expect M(r) → MG, with MG being the mass of our galaxy. If we continue to move outwards past this point, we see from the expression above that we expect the velocity to fall off as ∼ r−1/2 . What was actually seen in observations was that the circular velocity seemed to flatten out at large radii, tending to a constant value [15, 16]. From this one can imagine that the mass has to continue to increase beyond what we can actually see. This observation led people to consider the possibility that there could be a halo of DM that extends far beyond the radius of the visible matter. By considering contributions from the added halo, as well as from the observed matter, one can obtain quite nice fits to observational data [17]. It also seems possible to match both a universal DM density profile [18] as well as a universal velocity profile [19] to observed data, speaking in favour of the DM hypothesis.
2.2.2 Weak lensing observations Further evidence can be gathered with the help of galactic clusters from a phe- nomenon known as gravitational lensing. Gravity does not only affect massive objects, it also changes the trajectory of light as it passes by, or through, volumes with significant gravitational potentials. This can give rise to distortions in images of bright objects, or even multiple phantoms of the same object, should there be a region with strong gravitational fields between the observer and the observed ob- ject [13, 14]. This effect on light can be accurately calculated using GR and under certain circumstances, that are very commonly satisfied in astrophysical scenarios, the general shape of the gravitational potential can be recreated from the distor- tions found in images of the observed objects [20]. These types of measurements are commonly referred to as weak lensing experiments or weak lensing observations. One of the stronger arguments in favour of DM over versions of modified gravity is based in part on weak lensing observations. Observations of 1E0657-558, more 2.2. Observational evidence 5 commonly referred to as the Bullet Cluster, provide an excellent opportunity to observe a system where a large part of the baryonic content is expected to be sep- arated from the accompanying DM [21]. The system consist of two galaxy clusters that have collided. Galaxies pass right through each other, behaving essentially as collisionless particles, while the intra-galactic plasma is slowed down. It is known that for clusters in general the stellar content make up roughly 1-2 % of the mass and the plasma make up roughly 5-15 % [22, 23]. The plasma can be observed using X-ray astronomy and, if no DM is present, one would expect the gravita- tional potential to overlap significantly with the areas of high plasma abundance. Using weak lensing, the gravitational potential was reconstructed and it was found that the two peaks in the potential were significantly spatially separate from the plasma [21]. This speaks heavily in favour of DM that would, due to its very weak interactions, also pass unhindered through the collision, tracing the trajectory of the galaxies.
2.2.3 The cosmic microwave background The cosmic microwave background (CMB) is the remnant radiation from a much younger universe. In the early Universe the temperature was high enough to sep- arate electrons and baryons and together with the photons they formed what can best be described as a fluid. As the temperature decreased it was possible for electrons to bind together with protons to form neutral hydrogen, this period is referred to as recombination, and particle interactions with photons died off dras- tically. The photons left over from after recombination is what today is observed as the CMB [24, 25]. Today it is very well described by a black-body spectrum with a corresponding temperature of TCMB = 2.726 K [26]. It was initially seen as isotropic, i.e., equal in all directions, but it was soon discovered that it did contain some very slight anisotropies that revealed structure on smaller scales within it [27]. Today these anisotropies are the target of intense studies as they provide an ex- cellent probe for cosmological parameters and thus help us test our theories about the Universe. The focus of these studies is the power spectrum of the CMB. The quantity Θ = ∆T/T is expanded in terms of spherical harmonics and one can analyse the different modes and the relation between them. Before recombina- tion oscillations in the photon-baryon fluid are expected to appear as the photon pressure competes against gravity in areas of gravitational collapse. As recombi- nation takes place and the photons no longer scatter, they are red-shifted while climbing out of the gravitational potential, altering their wavelengths, giving rise to some of the anisotropies we see in the spectrum today. These are called primary anisotropies. Secondary anisotropies arise due to gravitational effects from pertur- bations to the metric, the mathematical object describing the shape of space-time in general relativity, as well as rescattering, i.e., some photons scatter again against objects during the later stages of the evolution of the Universe [24, 25]. Working with the so called ΛCDM cosmological model, measurements from the CMB lead us to believe that roughly 5 % of the energy density in the Universe 6 Chapter 2. Background is baryonic matter, while the total matter density is roughly 31 % [28]. Thus DM should make up approximately 26 % of the energy density of the Universe, or roughly 80 % of all matter. The rest of the energy density is attributed to the so called cosmological constant, Λ, which is interpreted as a “vacuum energy”. This parameter is responsible for the accelerating expansion of the Universe [20].
2.3 Alternatives to dark matter
The different observed phenomena covered above, as well as numerical simulations of structure formation, paint a picture where DM is a very attractive description. It is however worth mentioning that some alternative descriptions have been sug- gested. One of the theories that appeared in contrast to DM is the theory of modified Newtonian dynamics (MOND) [29]. This theory was developed mainly to describe observations of galactic rotation curves without introducing any DM. It is based on the idea that Newton’s second law should be modified to take the form
F = mµ(a/a0)a. (2.2) Here µ(x) is a function with the limiting values
µ(x) ≈ 1 for x 1, µ(x) ≈ x for x 1. (2.3)
The constant a0 can then be chosen such that the second limit is valid for stars far out from the hosting galaxy’s centre and this would in turn yield a constant circular velocity that depends on the mass of the galaxy and a0. There exists a generalisation of the MOND concept to a relativistic setting, referred to as tensor- vector-scalar (TeVeS) gravity [30]. A large issue for both of these theories is the fact that they cannot describe the phenomenon observed in the Bullet Cluster mentioned in subsection 2.2.2. Another interesting issue for MOND is the very recent discovery of a galaxy that may lack DM [31]. Since the existence of ordinary matter on sufficient scales would always give rise to DM like behaviour in MOND, the fact that baryonic matter and DM could exist as completely separate objects is not consistent with MOND-based theories.
2.4 Structure issues of dark matter
Even though the evidence in favour of cold collisionless DM is plenty, there are still some issues present in this picture. A very short introduction to three of the more famous ones is presented below.
2.4.1 Missing satellites The missing satellites problem is related to the number of DM subhalos present around the Milky Way DM halo. Numerical simulations of the formation of DM 2.4. Structure issues of dark matter 7 halos on the scale of those hosting galaxies and galactic clusters predict the existence of a significant number of smaller subhalos surrounding the main one. However the number of satellites found in numerical simulations greatly outnumber that of the actually observed subhalos around the Milky Way [32–34]. There have been discussions of how to alleviate this issue and recently a new numerical simulation was carried out that took into account a number of different baryonic effects [35]. They conclude that no missing satellites appear in their results. Until this has been further verified it is unclear whether this issue is resolved or not.
2.4.2 Cusp versus core When looking into the actual structure of the subhalos and the smaller galaxies within them another discrepancy between observations and numerical simulations appears. This one is concerned with the behaviour of the DM density as a function of radial distance from the galactic centre. Numerical results seem to favour a so called “cusp” region close to the central parts, where the density diverges [18, 34, 36]. Observations are however in disagreement, with results instead tending to a so called “core” region where the density flattens out closer to the centre [37–40].
2.4.3 Too big to fail Based on the simulations in Ref. [34], one can compare features of the predicted subhalos with those actually observed around the Milky Way. Specific comparisons of the halos around the Milky Way that host dwarf galaxies were made and it was noticed that the numerics predicted the existence of several halos larger than these. The question then arose, if these actually do exist, why are they not hosts to galaxies, just as the smaller subhalos are? Their size makes them good candidates for galaxy formation according to standard models of how galaxies form [41, 42]. A number of suggestions have been put forward to solve these issues and one such suggestion is to increase the DM interactions sufficiently on the relatively small scales, where these issues are prominent, while leaving the behaviour on larger scales intact as they are. Increasing the interaction cross section on the relevant scales may turn cusps into cores, alleviate the “too big to fail” problem by reducing the velocity dispersions of larger subhalos, and make subhalos more susceptible to disruptive events and thereby possibly alleviating the missing satellites problem [43].
Chapter 3
General theory
3.1 Inelastic dark matter
Inelastic DM is a class of models that were originally introduced in order to resolve a discrepancy between two major direct detection experiments [10]. The DAMA experiment claimed to, and still do, observe a signal consistent with predictions for a WIMP [8, 44]. However, the entire parameter space of the signal was quickly ruled out by the CDMS experiment [9]. There are a number of other direct detection experiments such as XENON [45], LUX [46], and PandaX [47] that are actively searching, but so far none of them have found anything and all of them rule out the DAMA signal. In the simplest form, an inelastic DM model consists of two DM states, χ1 and χ2, with masses m1 and m2, respectively. The masses are nearly equal, but a small difference δ exists such that
m2 − m1 = δ, |δ| m1, m2. (3.1) An example of a Lagrangian for such a model is [48]
1 0 2 1 0 0µ X 1 µ L = − F − m 0 A A + χ¯ (iγ ∂ − m ) χ 4 µν 2 A µ 2 i µ i i i=1,2 (3.2) ig ig + χ¯ γµχ A0 − χ¯ γµχ A0 . 2 2 1 µ 2 1 2 µ
0 Here χ1 and χ2 are the DM fields and A is a gauge boson field with mass mA0 . The two last terms are the interaction terms and it can be seen that they always connect the two different DM states, hence we obtain “inelastic” DM. The basic idea of why such a model resolves the discrepancies mentioned above is illustrated with the following example. Imagine the two DM particles, χ1 and χ2
9 10 Chapter 3. General theory
with masses m1 and m2 as above. We now look at the scenario of χ1N scattering, where N is some target particle and let w be the relative speed between the DM particle and the target. Due to conservation of energy we obtain the following criterion for the scattering to be possible
w2m m δ ≤ 1 N . (3.3) 2(m1 + mN ) Thus, the occurrence of scattering events depends on the mass of our target particle, as well as the DM mass difference δ. It could then be the case that scattering events would take place in the DAMA experiment, which used NaI molecules as their target, but not in CDMS, which used Ge atoms. This due to the lower mass of Ge compared to I [10]. So by considering an inelastic scattering process, it was considered plausible to reconcile the two. Today it is rather difficult to try and explain the results of DAMA in contrast to the null results of other experiments with inelastic DM [49]. Before moving on to capture and scattering of DM in the Sun, a bit of prepara- tory work is needed. A lot of the material in the following sections are reproductions of results from Refs. [11, 50]. They are covered here for completeness.
3.2 Kinematics
Consider again the case of two DM particles, χ2 and χ1, such that m2 − m1 = δ with |δ| m1, m2, with only inelastic scattering taking place. Assume now that we wish endothermic scattering of the χ1 to take place against some target particle N with mass mN . In an endothermic event energy is absorbed in the process, this in contrast to exothermic processes which release energy. The total energy of the χ1-N system before the scattering event is given, in the centre of momentum frame, by 1 E = µ w2. (3.4) 2 1
Here we introduce the relative velocity w between the χ1 and the target as well as the χ1-N reduced mass µ1, defined as per usual µ1 = m1mN /(m1 + mN ). For scattering to take place we need to at least overcome the initial energy barrier of the mass difference between χ1 and χ2. We thus obtain a first basic kinematic constraint for scattering to be possible, namely eq. (3.3). Note that, since the velocities we will typically deal with are far below the relativistic limit [12, 13, 51], we only need to worry about Galilean transformations rather than Lorentz transformations. The next thing we would like to figure out is the recoil energy of the target particle, something that is only well defined in the rest frame of the target. We will, however, compute it by finding the target particle’s velocity in the centre of momentum frame and then perform the relevant transformation to the rest frame. 3.2. Kinematics 11
We will keep track of “equivalent” quantities in the target rest frame and centre of momentum frame by denoting them with or without a 0, respectively. Due to conservation of energy we obtain, using a non-relativistic expansion of the energy- momentum relation,
1 2 0 0 m1 + mN + µ1w = m2 + mN + E + E , (3.5) 2 χ2 R where the left-hand side refers to before the scattering and the right-hand side to after the scattering event. Furthermore, E0 and E0 are the χ and target particle χ2 R 2 kinetic energies, respectively. Since we work in the classical regime and in the centre of momentum frame, we can easily relate the kinetic energies to the magnitude of the velocity of N, which we denote u0. We find that
0 2 µ2 2 (u ) = µ1w − 2δ , (3.6) mN 0 where we have introduced µi = mimN /(mi + mN ). If we define an angle θ from the collision axis, we can write down the two-vector form of the target particle’s velocity
r 0 0 µ2 2 cos(θ ) u = (µ1w − 2δ) 0 . (3.7) mN sin(θ ) It is now a simple task to transform this result to the initial rest frame of N, where we find the recoil velocity q µ2 2 0 µ1 (µ1w − 2δ) cos(θ ) + w mN mN u = q (3.8) µ2 2 0 (µ1w − 2δ) sin(θ ) mN and from this we can compute the recoil energy
s 2 2 " r # 0 µ1w µ2 µ2 2δ 0 µ1δ ER(w, θ ) = 1 + + 2 1 − 2 cos(θ ) − . (3.9) 2mN µ1 µ1 µ1w mN
We can thus obtain the maximum and minimum recoil energies possible for a given relative velocity w by setting cos(θ0) = ±1, respectively. We obtain the expressions
s 2 2 " r # max(min) µ1w µ2 + µ2 2δ µ1δ ER (w) = 1 + (−) 2 1 − 2 − . (3.10) 2mN µ1 µ1 µ1w mN
Due to the very minor errors induced relative to the computational ease it provides, we will often work with the slightly simpler case of µ2 = µ1 = µ = m1mN /(m1 + mN ) when performing actual computations. 12 Chapter 3. General theory
One other quantity that we are going to need for later use is the angle θ between the velocities of the incoming χ1 and the outgoing χ2 particle in the initial rest frame of the target particle. We can obtain this from simple energy and momentum conservation. We define the kinetic energies of χ1, χ2, and N to be E1, E2, and EN . The energies are related to the momenta of the particles by the classical relations
2 |pi| Ei = , i = 1, 2,N. (3.11) 2mi Momentum conservation now leads to
pN = p1 − p2. (3.12) Squaring both sides and substituting in the energy expressions instead, we find that p p 2mN ER = 2m1E1 + 2m2E2 − 2 2m1E1 2m2E2 cos(θ). (3.13) Solving for (the cosine of) our angle we obtain m E + m E − m E cos(θ) = 1 1 √ 2 2 N R . (3.14) 2 m1m2E1E2 3.3 Orbital mechanics
We will now spend some time on developing relations relating to the orbital me- chanics of particles. Consider the effective Hamiltonian for a particle in a central potential
1 J 2 H = mr˙2 + + V (r), (3.15) 2 2mr2 where J is the angular momentum of the particle and V (r) is the potential energy. As mass is a parameter that we may vary, it is convenient to define “reduced” quantities where the mass has been factored out. Let us therefore define the reduced angular momentum L through the relation
J ≡ mL (3.16) and a gravitational potential φ(r) through
V (r) ≡ mφ(r). (3.17) Substituting both of these into the effective Hamiltonian we may also define the reduced energy E according to
1 L2 1 H = m r˙2 + + φ(r) = m r˙2 + φ (L, r) ≡ mE, (3.18) 2 2r2 2 eff where we also define the effective potential 3.3. Orbital mechanics 13
L2 φ (L, r) = + φ(r). (3.19) eff 2r2 When speaking of energy and angular momentum from this point onwards it is always these reduced quantities that we refer to unless otherwise stated. For a given bound state energy E < 0 there is a range of 0 < L < Lmax(E) that defines different orbits. We can confine the parameter space a bit. We exclude all orbits that never pass through the Sun, i.e., orbits that always satisfy r > R . We can then find the maximum angular momentum for a given energy as the L satisfying
min (φeff (L, r)) , r < R . (3.20)
Consider now a χ1 particle that very far away from the influence of the potential source have the velocity v. We need to express the velocity a distance r from the centre of the potential. We know from conservation of energy that
1 1 m v2 = m w2(r) + m φ(r) (3.21) 2 1 2 1 1 and we can thus solve for w(r) and find
2 2 2 2 w (r) = v − 2φ(r) ≡ v + vesc(r), (3.22) p where we defined the escape velocity vesc(r) = −2φ(r). We now wish to figure out how the quantities E and L change as a particle scatters, considering the case where the initial values are given. The energy of the outgoing particle is found from energy conservation as soon as the recoil energy of the scattering event is determined. The outgoing particle energy is given by
δ Eo = Ei − ER − , (3.23) m1 with Ei being the energy of the DM particle before scattering. Since we know the energy we also know the outgoing velocity of the particle. What needs a bit more consideration is the angular momentum of the outgoing particle. We here refer specifically to the energy and angular momentum of the particle in the Sun’s rest frame. Angular momentum is given by
L = r × w (3.24) and its magnitude can be found using the familiar relation for the magnitude of a vector product
L = rw sin(α), (3.25) where α is the angle between r and w. The square of the magnitude of the angular momentum is what we will need most of the time and it is given by 14 Chapter 3. General theory
L2 = r2w2 sin2(α) = r2w2 − (r · w)2 . (3.26) Thus, what we really need to find is the velocity vector of the particle in the solar frame after the scattering event. In order to do this, first consider a coordinate system that has its x-axis aligned with the incoming particle’s velocity vector wi and let the angle between the incoming velocity vector and the outgoing velocity wo be θ. We already know how to compute (the cosine of) θ, i.e., by plugging in our energies into eq. (3.14). There is also an arbitrary angle, call it ϕ, with which the outgoing velocity may be rotated around an axis aligned with the incoming velocity. In this coordinate system, our outgoing velocity is then given by
cos(θ) wo = wo sin(θ) cos(ϕ) . (3.27) sin(θ) sin(ϕ) Now we need to be able to compute the inner product in eq. (3.26). Let us thus make a change of coordinates to a system where r is aligned with the x-axis, and wi is located in the x-y plane at an angle α from r. This transformation can be carried out using a rotation matrix
cos(α) − sin(α) 0 R(α) = sin(α) cos(α) 0 . (3.28) 0 0 1 In this frame our outgoing velocity vector is given by
cos(α) cos(θ) − sin(α) sin(θ) cos(ϕ) wo = wo sin(α) cos(θ) − cos(α) sin(θ) cos(ϕ) (3.29) sin(θ) sin(ϕ) and we obtain the outgoing angular momentum
2 2 2 Lo = r wo − (r · wo) s 2 L 2 L (3.30) = r2w2 1 − 1 − i cos(θ) − i sin(θ) cos(ϕ) , o rw rw i i where use has been made of the relation sin(α) = Li/(rwi). We thus know how to obtain the energy and angular momentum after a scattering event has taken place, given the initial values, a certain recoil energy, and a number of random collision parameters. These relations will be needed later to properly keep track of and update the energy and angular momentum of the particles in our simulation. Next, we can find the maximum and minimum radii for a given value of E and L. This is done by noting thatr ˙ = 0 at these extrema. Plugging this into eq. (3.18) we find that 3.4. Differential scattering cross section 15
L2 E = 2 + φ(rmin(max)), (3.31) 2rmin(max) which can be solved to give the relevant radii when the exact form of φ(r) is specified. We can also find the time that a particle spends between two radii for a given E and L by first isolatingr ˙ in eq. (3.18) and then solving the separable differential equation obtained. Doing this we find that
Z t2 Z r2 1 T (r1, r2) ≡ dt = p dr . (3.32) t1 r1 2 [E − φeff (L, r)]
3.4 Differential scattering cross section
A quantity of large importance is the scattering cross section σ or, in this case, the differential scattering cross section dσ / d|q|2. It tells us, in rough terms, how likely a scattering event is to occur as we shoot particles at each other. Often the scattering cross section is described using two parts, the zero-momentum transfer cross section σ0 and a form factor F (|q|) defined such that F (0) = 1. The form factor takes the internal structure of our target into account, since we are not scattering off of point particles. Depending on the interaction type in our scattering event the computation of the form factor may be very complicated, but it can be thought of as the Fourier transform of our target mass distribution [12]. The form we will use for the differential cross section is
dσ σ0 2 2 = 2 2 |F (|q|)| . (3.33) d|q| 4µ1w
Here µ1 is the reduced mass defined earlier, w the relative velocity of the incoming DM particle and the target, and q is the momentum transfer during the scattering event [12]. The momentum transfer is related to the recoil energy of the target particle through the relation
p |q| = 2mN Er. (3.34)
Using this relation we can make a change of variables to obtain the differential cross section in terms of the recoil energy instead
dσ mN σ0 2 = 2 2 |F (ER)| . (3.35) dER 2µ1w The explicit form of the extra factors present in the differential cross section is due to normalisation, which is chosen such that 16 Chapter 3. General theory
Z Emax dσ dER = σ0. (3.36) dE Emin R ER=0 Plugging in the elastic case into eq. (3.10) we thus obtain the normalisation condi- tion
2µ2w2 Z m N dσ dER = σ0 (3.37) dE 0 R ER=0 that precisely yields the normalisation factor present in eq. (3.35). We would now like to adapt this expression a bit for use in the inelastic case. The normalisation must be adapted for this scenario and we instead find that
dσ σ0 2 = |F (ER)| , (3.38) dER ∆E where ∆E ≡ Emax(w) − Emin(w), with the energy limits given by eq. (3.10). The zero-momentum transfer cross section σ0 will also have to be modified slightly. This due to a factor of |pi|/|pf |, which in the elastic case turns out to be 1, present when computing cross sections [52]. Here, pi and pf refers to one initial and final momenta involved in the scattering event in the centre of momentum frame. Here we use the initial and final momenta of the target particle. The initial momentum is |pi| = µ1w (3.39) and the final momentum is found to be
r s µ2 2δ |pf | = µ1w 1 − 2 . (3.40) µ1 µ1w Thus, the sought fraction turns out to be
r s |pi| µ2 2δ mN ∆E = 1 − 2 = 2 2 (3.41) |pf | µ1 µ1w 2µ1w and our modified zero-momentum transfer cross section is
mN ∆E σ0,inel = 2 2 σ0. (3.42) 2µ1w Putting the pieces together we arrive at a differential cross section for our inelastic process of the form
dσinel mN ∆E 1 mN σ0 2 = 2 2 σ0 = 2 2 |F (ER)| , (3.43) dER 2µ1w ∆E 2µ1w which is identical to eq. (3.35). Any possible peculiar extra factors cancel and we will thus continue to refer to this quantity simply as dσ / dER. We will take our cross section σ0 to be the regularly used spin-independent one, given by [10, 11] 3.4. Differential scattering cross section 17
µ2(f Z + f (A − Z))2 σ = 1 p n σ . (3.44) 0 f 2µ2 χ1p n χ1p Here A is the total number of nucleons in our target nucleus, Z the number of protons, σχ1p the DM-proton cross section, and fp (fn) represents the interaction strength between DM and protons (neutrons). We will work with the case of fn = fp and consider the case of equal interactions between DM and the two types of nucleons. We thus end up with the following expression for the differential cross section
dσ m A2σ = N χ1p |F (E )|2. (3.45) dE 2µ2 w2 R R χ1p Involving a light mediator particle, denoted by A0, in the process somewhat com- plicates the expression a bit. It introduces a long-range interaction where the corresponding quantity to σχ1p now contains a momentum transfer dependence of the following form [53, 54]
2 2 −2 σχ1p(q) ∼ q + mA0 . (3.46) In the limit when the mediator mass is very heavy, eq. (3.35) is recovered [53]. In an attempt to isolate this behaviour, and to simplify the comparison with earlier results later on, we define a “reference cross section” σR ≡ σχ1p(qR) for a specific reference momentum transfer qR. In the general case we would then have
2 2 2 qR + mA0 σχ p(q) = σR. (3.47) 1 2 2 2 (q + mA0 ) We could then incorporate this new behaviour into our earlier expression for the cross section and write
2 2 2 2 dσ mN A σR qR + mA0 2 = 2 2 2 |F (ER)| . (3.48) dER 2µ w 2 2 χ1p (q + mA0 ) The total cross section is then given by
Z Emax dσ σ = dER . (3.49) Emin dER Parametrising our cross section in this way also ensures that we fall back to eq. (3.45)
0 when taking the limit of large mA , where σR plays the role of σχ1p. This cross section would in general decrease as our particles move faster and transfer more momentum in collisions. Precisely the behaviour needed to possibly alleviate the issues reviewed in section 2.4 and one of the reasons for light mediators to be introduced [43]. Reasonable cross sections to solve these issues have been achieved in the setting of inelastic DM [55]. 18 Chapter 3. General theory
3.5 Decay of the χ2 state
We will here consider the case where mA0 < δ. This would enable the heavier DM 0 state χ2 to decay into the lighter DM state χ1 and the boson A . We now wish to compute the lifetime of the heavier state. The exact model used is not critical, but for our simulations later to be compu- tationally viable we would like the lifetime of χ2 to be relatively short. “Short” is here referring to the fact that we would like our heavier state to decay well before there is any significant risk for it to scatter or travel a noticeable distance. In the simulations a solar model, specified in section 5.1, will be used, where the Sun is discretised into radial shells of thickness ∆r = 0.0005R . We would like the life- time of χ2 to be shorter than the time it takes for the heavy state to pass through c one such shell. Using the escape velocity from the Sun’s core, vesc ≈ 1381 km/s [56] c to estimate this time we end up with a value of tShell = 0.0005R /vesc ≈ 0.25 s. Note that this is a lower limit of the time it may take for a gravitationally bound particle to cross such a shell. We use the interaction terms present in eq. (3.2) to compute an example life time. The interaction terms can be rewritten into a single equivalent term
µ 0 LInt = igχ¯2γ χ1Aµ. (3.50) The Lagrangian used is then of roughly the same form as found in, for example, Ref. [57]. The decay rate of χ2 due to this interaction term can now be computed, and the following expression is obtained in the centre of momentum frame
¯ 2 q0 M CoM ΓCoM = 2 , (3.51) 8πm2 where
q 2 2 2 2 2 2 (m1 + m2 − mA0 ) − 4m1m2 q0 = (3.52) 2m2 and
2 2 (δ − m 0 )(δ + m 0 ) (2m + δ) + 2m 0 ¯ 2 2 A A 1 A M CoM = g 2 . (3.53) mA0 −1 The expression for the lifetime of χ2 is τCoM = ΓCoM. The lifetime as a function of the dark coupling constant g is shown in Figure 3.1. It does not seem unreasonable to expect the heavier state to have the required short lifetime, or lower. When the involved masses are fixed the kinematics of the decay can be determined. The resulting χ1 particle will emerge with a velocity given by 3.5. Decay of the χ2 state 19
Figure 3.1: Lifetime of the heavier DM state in its rest frame as a function of the coupling constant g.
s 2 2 δ − m 0 u = A (3.54) m1m2 in a random direction in the χ2 rest frame.
Chapter 4
Dark matter in the Sun
The Milky Way, like most other galaxies, is thought to be embedded in a large DM halo. If this is the case, then one would expect the Solar System to experience a flux or “wind” of DM to pass through it due to the rotation of the Milky Way. If a DM particle was to scatter while passing through the Sun, it may lose a sufficient amount of energy to become gravitationally bound to the Sun. Subsequent scattering events are then unavoidable until it either gains enough energy in a scattering event to break free of the gravitational pull or encounters some other kinematic constraint that prevent further scattering. The topic of DM in the Sun has been the subject of discussions for quite some time [58, 59]. The interest is due to any observable effects the DM may have on the Sun and the Sun is being used as a DM laboratory for indirect detection. The focus is mainly on neutrinos resulting from possible DM annihilation within the Sun and several searches for these are ongoing [60– 62]. These neutrinos would have a much higher energy compared to regular solar neutrinos and their detection would be a major step towards pinning down DM. As indirectly observable effects of DM in the Sun would be due to DM annihila- tion, we need to figure out the corresponding annihilation rate. The number of DM particles in the Sun at a given time t is given by N(t) that satisfies the differential equation [11, 63]
2 N˙ (t) = C − CevapN − CannN . (4.1) Here, C is the capture rate of DM due to scattering on normal matter in the Sun and we do not expect it to depend on the number of DM particles already captured, CevapN is the evaporation rate of DM from the Sun which we expect to be proportional to the number of DM particles already present. Finally we have 2 CannN that describes the rate of DM-DM annihilation. We expect this term to be proportional to the square of the number of DM particles present. This can be seen qualitatively from the following argument: If N particles are present, then there are N(N − 1)/2 pairs of DM particles that can annihilate. As N grows large,
21 22 Chapter 4. Dark matter in the Sun the quadratic term will dominate and the factor of 2 is cancelled by the fact that a single annihilation event destroys two DM particles. Equation (4.1) has the solution
C tanh t N(t) = τ , (4.2) −1 Cevap t τ + 2 tanh τ where the characteristic time τ is given by
" #−1/2 C 2 τ = C C + evap . (4.3) ann 2
If we disregard evaporation we can see from eq. (4.1) that we reach a steady state when the contributions due to capture and annihilation balance out. Thus, if a steady state is reached, observables due to DM annihilation may instead be characterised by the capture rate.
4.1 Solar capture
The computation of the capture rate of DM inside an object is considered to be a standard calculation and dates back quite a while. A reproduction of the compu- tation found in Ref. [7] will be presented here. Consider DM distributed in the local neighbourhood according to some velocity distribution f(v) and consider further a sphere with a radius R large enough that the DM have yet to become influenced by the gravitational potential of the Sun. The fraction of the DM particles that pass through this sphere in an angular interval [θ, θ + dθ] from the radial direction is given by
dΩ 2π sin(θ) dθ 1 = = sin(θ) dθ . (4.4) 4π 4π 2 The flux of particles with quantities in the range [θ, θ + dθ] and [v, v + dv] is then
1 1 sin(θ)n f(v)v cos(θ) dθ dv = − n f(v)v dv d cos2(θ) . (4.5) 2 χ 4 χ
Note that the local galactic DM number density nχ has here been explicitly fac- tored out from the distribution function. This corresponds to the use of the same normalization as used in Ref. [11], where Z Z f˜(v) d3v = f(v) dv = 1. (4.6)
We now need to figure how how much time a particle will spend within a thin shell at a specific radius r < R. Here, the particle will have a velocity w given by eq. (3.22) and thus the time it takes to pass through the shell is 4.1. Solar capture 23
dr dr dτ0 = = q , (4.7) w cos(θ) L 2 w 1 − rw where eq. (3.25) has been used. Due to conservation of both energy and angular momentum, a particle will either pass through the shell twice or not at all, and its angular momentum is restricted to 0 ≤ L ≤ rw. We thus obtain the following expression for the total time spent in the shell
dτ = 2Θ(rw − L) dτ0 , (4.8) where Θ(x) is the Heaviside step function. We can now also make use of eq. (3.25), evaluated at R, to rewrite our expression for the flux as
1 1 f(v) − n f(v)v dv d cos2(θ) = n dv dL2 . (4.9) 4 χ 4 χ vR2 The total flux over the entire large spherical surface is then
1 f(v) f(v) 4πR2 dv dL2 = πn dv dL2 . (4.10) 4 vR2 χ v We now denote the scattering rate into a bound state of a DM particle with veloc- ity w as S(w). At a radius r, the probability of scattering into a bound state is then
Pbound = 2S(w) dτ0 Θ(rw − L). (4.11) With the scattering cross section σ from eq. (3.49), a DM particle velocity w, and a target particle number density nN (r), we would have
Z Emax dσ S(w) = nN σw = nN w dER , (4.12) Emin dER yielding the differential expression dσ dS = nN w dER . (4.13) dER Putting all these pieces together, we obtain an expression for the differential capture rate
f(v) dσ 1 2 dC = 2πnχ nN (r)q dr dv dER dL . (4.14) v dER L 2 1 − rw Note that the Heaviside step function has been removed from this expression, since it only affect the integration limits. This expression does not take into account any effects that may occur due to the additional velocity contribution from the decay of the heavier DM state. This 24 Chapter 4. Dark matter in the Sun contribution will be neglected here as it is very small compared to the velocities typically involved in the capture process.
4.2 Scattering in the Sun
As mentioned earlier, when a particle has been captured into a gravitationally bounds state it has no other option but to remain bound and subsequently scatter again until it evaporates or cannot scatter further due to kinematic constraints. The rate at which DM particles are scattered in the Sun can be computed in a rather straight forward fashion [11]. Assuming that the target particles in the Sun are described by a velocity distribution fN (r, v) and that the velocity of the DM particle is given by w we have the differential scattering rate
3 dΓ = σnN (r)fN (r, v)|w − v| d v , (4.15) where nN (r) is the number density of our target particle at radius r and σ is given by eq. (3.49). The integral here must be calculated in the rest frame of the target particle where the recoil energy is well defined. The expression above in principle gives us the total scattering rate at a specific radius or, if we were to integrate over all radii, the entirety of the Sun. We can however use it to compute the scattering rates amongst different points in our E-L phase space. Also, as with the capture rate, we must explicitly add effects related to the heavier DM state decaying when computing these scattering rates. How both of these things are done is described in section 5.1.
4.3 Dark matter distribution in the Sun
As mentioned above, the quantity of interest is the annihilation rate of DM in the Sun. To compute it one would in general need to know how the DM is distributed in the Sun. If the DM particles are described by a number distribution function n(r, v) = Nf(r, v), we then get the DM-DM annihilation rate
Z 2 3 3 3 2 A = N σ|v1 − v2|f(r, v1)f(r, v2) d v1 d v2 d r = CannN , (4.16) with N being the total number of DM particles in the Sun, and v1 and v1 are the velocities of the two annihilating DM particles. We arrive precisely at the form discussed in relation to eq. (4.1). It is apparent from the expression above that the form of the DM distribution function may have a large influence on the annihilation rate. If DM interact rather rarely, i.e., it completes several orbits in between scatter- ing events, a common assumption is that it can be described using an isothermal Maxwell-Boltzmann distribution [64–67]. Such a distribution is given by 4.3. Dark matter distribution in the Sun 25
E(r, v) fIso(r, v) = C exp − , (4.17) kBT with C being a normalisation constant, E(r, v) the energy of our particles, kB the Boltzmann constant, and T the temperature of the distribution. If we wish to obtain the radial distribution we begin by integrating out the kinetic part, absorbing the result into C and we then end up with m1φ(r) fIso(r) = C exp − . (4.18) kBT The normalisation condition is that we obtain 1 when integrating over the volume of the Sun Z 3 fIso(r) d r = 1. (4.19) V Assuming further a constant density and an isotropic distribution we can write it as
r2 f (r) = Cr2 exp − , (4.20) Iso r∗2 with G being the gravitational constant, and r∗ a characteristic length given by ∗ p r = 3kBT/(2πGρm1). For a DM particle mass of m1 = 100 GeV and both temperature and density corresponding to the Sun’s core, the characteristic length ∗ is roughly r ≈ 0.01R . This allows us to replace the upper integration bound when normalizing from R to infinity without introducing any significant error. It also verifies that our assumption of constant temperature and density is valid, since the DM is mainly confined to such a small region. Normalising our distribution we obtain
4r2 r2 f (r) = √ exp − . (4.21) Iso πr∗3 r∗2 While this has been shown to be a good approximation for elastic DM [68], it has also been shown to be a very bad one for inelastic DM where both DM states are stable and the scattering events are point like [11].
Chapter 5
Numerical simulations and results
5.1 Numerical setup
In order to carry out numerical simulations of the capture and scattering rates, we discretise the quantities E and L uniformly into 100 states in the intervals [Emin,Emax] and [0,Lmax(E)], respectively. This results in a total of 10000 states (E,L). A particle in the simulation is then characterised by these two quantities. The equation governing the time evolution of the discretised states is then given by [11]
˙ X X fα = Cα + Σαβfβ − fα Aαβfβ. (5.1) β β
Here, fα describes the number of particles in a particular state α, that is charac- terised by a unique combination (E,L), Cα is the capture rate into that state, Σαβ contains the scattering rate for particles to transition from state β to α, and Aαβ governs the annihilation of a state β with another state α. To compute Cα eq. (4.14) is discretised in the four involved variables. For 2 each discretisation point (ri, vi,ER,i,Li ), we can compute the energy, speed, and in-plane scattering angle of the particle post collision with the help of eqs. (3.22), (3.23), and (3.14). Using random values for the angle ϕ in eq. (3.30) we can find out how likely it is for a particle in each discretisation point to end up in a particular captured state α. The capture rate Cα is then given by summing over all the contributions from each discretisation point multiplied by the probability for the particle to end up in a state α. To compute the scattering matrix Σ we proceed in the following manner. For each α we choose the corresponding E and L as initial values. From here we generate a random collision event at some radius ri by picking a random target particle velocity, making sure that it leads to a valid scattering
27 28 Chapter 5. Numerical simulations and results event, and a random recoil energy. From this we can compute the outgoing velocity vector of our DM particle, and here we add a velocity contribution due to the decay in a random direction. We then compute the final state β our DM particle would end up in, given this particular scattering event. We could now, in principle, compute a, in general rather poor, one sample Monte Carlo estimate of the integral of eq. (4.15). Since we know both the initial and final state of our scattered particle, this actually gives us an estimate of Γα→β(ri). We now instead generate a very large number of randomized scattering events, keeping track of the resulting final states, from which we should be able to get a good Monte Carlo estimate of Γα→β(ri), as we now average over a larger number of samples for each final state. The possible radii for our collisions are specified by the discretised radii available in the AGSS09ph solar model [69]. The off-diagonal elements of the scattering matrix are then given by summing over all contributions from each radius, weighted by the fractional time Tα(ri) a particle in state α spends at that radius, X Σβα = Γα→β(ri)Tα(ri). (5.2) i
The fractional time for a specific state α and radius ri is defined by
T (rinner, router) Tα(ri) = , (5.3) T (rmax, rmin) where rinner and router are here the inner and outer radii of a spherical shell centred on the radius ri. The diagonal elements of the scattering matrix are given by X Σαα = − Γα(ri) (5.4) i to account for all the particles that scatter away from the state α. Note that we have accounted for the decay of the heavier state when we compute the scattering matrix, but not when computing the capture rate into each state. Neglecting the extra velocity contribution due to the decay when computing the capture rate will be a very good approximation as the velocities involved in the initial capture is always greater than ∼ 10−3. This can be seen, for example, by computing the escape velocity at the solar surface which is found to be ∼ 0.002. The velocity is generally not reduced by much due to the scattering event, resulting in the captured particles having a similar velocity. The contribution due to the decay is, roughly, −6 of order ∼ δ/m1 = 10 , which is far smaller and will thus not have any noticeable effect. However it is not as apparent that this approximation would work as well when computing the scattering rate. The velocities resulting from scattering events with bound state DM may be much lower. 3 In these simulations the local DM density is taken to be nχ1 = 0.4 GeV/cm [70, 71]. Due to the heavier state decaying, we only consider χ1 to be present in the DM halo. We use the spin-independent cross section defined in eq. (3.48) with the values −42 2 σR = 10 cm and qR = 50 MeV throughout this thesis. The reference value of 5.1. Numerical setup 29
the momentum transfer corresponds roughly to a recoil energy of ER = 10 keV in direct detection experiments using xenon as target particles [45–47, 72]. The Helm form factor [73, 74]
3j (qR) −q2s2 F (q) = 1 exp (5.5) qR 2 is used, where
r 7 R = c2 + π2a2 − 5s2, (5.6) 3 1/3 c = 1.23A − 0.6 fm, a = 0.52 fm, s = 0.9 fm, and j1(x) is the spherical Bessel function of the first kind with l = 1 and is given by
sin(x) cos(x) j (x) = − . (5.7) 1 x2 x The velocity distribution for DM in the galaxy is taken to be the same as in Refs. [11, 75]
2 2 v 3(v − v ) 3(v + v ) f(v) = 2 exp − 2 − exp − 2 , (5.8) πv 2vrms 2vrms with the Sun’s velocity through the galaxy v = 220 km/s and the velocity dis- persion vrms = 270 km/s. In order to obtain a good description of the Sun’s com- position, the AGSS09ph solar model [69] is used. Not all elements available in the model are used in the simulations. The elements used are hydrogen, helium, nitro- gen, oxygen, neon, and iron since these are the elements that contribute the most to the scattering rates [11]. The same DM mass is used in all simulations and it is set to m1 = 100 GeV. Mainly two different mediator masses were considered, these are mA0 = 20 keV and mA0 = 40 keV. These are both very small compared to the expected momentum transfers in the scattering event, the minimum recoil energies given by eq. (3.10) correspond to momentum transfers of the order ∼ 10 MeV and they are both smaller than the narrowest mass difference considered, which was δ = 50 keV. Some exceptions to these two mediator masses will be considered. In these scenarios one might imagine that two different particles, except for the DM states, are involved, one in the decay process for DM and another in the scattering events. This allows us to move away from the restrictions due to δ when consider- ing scattering events and we may study how increasing the relevant mediator mass affects the capture and scattering rates. To keep these scenarios separated we will denote the lighter particle taking part in DM decay by A0 and any other particle that may act as a mediator, but which does not take part in the decay, by B0. As noted in section 3.4 we expect to recover earlier results without a light mediator if we take the mass of the mediator involved in the scattering to be much larger than the momentum transfer. This is an important step in verifying that our numerics behave as expected and agree with earlier results, mostly with those of Ref. [11]. 30 Chapter 5. Numerical simulations and results
When evolving our captured distributions in time we will consider the case where an initial set is captured and then evolved, with no further capture taking place during the evolution process. Annihilation of DM will also be neglected. The distribution at a time t is then given by [11]
f(t) = eΣtf(0). (5.9) We can transform the distribution in E-L space back to a radial distribution. This is done by weighting the number of particles in a particular E-L state by the fractional time that state spends at each radius X f(r) = fαTα(r). (5.10) α
5.2 Solar capture rate
Using the setup described above we compute the capture rate. The resulting total capture rate into different states is shown in Figure 5.1 for m1 = 100 GeV, mB = 40 keV, and δ = 50, 100, 150, 200 keV, respectively. It can be seen that, in general, the DM seems to be captured into somewhat loosely bound states rather than tightly bound ones. This can be traced back to the fact that larger recoil energies are highly suppressed, mainly due to the mediator’s contribution to the cross section and, to a lesser degree, due to the form factor. It is also seen that DM particles become more tightly bound when δ increases, which is very reasonable. As we increase the mass difference between the two DM states a larger portion of the χ1 particle’s energy is required to create the χ2 state, leaving the resulting particle with less total energy. The majority of the capture rate is due to the heavier target elements and neither hydrogen or helium contributes at all. The reason for this can be argued from eq. (3.3). The required velocity for a scattering event to be possible at all is
2δ w2 > . (5.11) µ1
For hydrogen and helium, which are much lighter than our DM particle, µ1 ≈ mN , while for iron we have µ1 ≈ m1/3. Thus, for the case of hydrogen or helium, the required velocity is many times larger than it is for iron. It is interesting to note the asymmetry in capture rate with respect to energy. The large capture rate “area” in our (E,L) phase space is much flatter below the high capture region than above it. This is thought to be a remnant from the distribution the DM particles follow before being captured in the Sun, which we took to be a Maxwellian distribution. Our mediator mass is much smaller than the momentum transfer in these scattering events and as such the cross section for the process in principle scales as ∼ q−4. Any deviation from the very smallest of momentum transfers would thus lead to significant suppressions in the capture 5.2. Solar capture rate 31
Figure 5.1: The computed total capture rate into the different E-L states for dif- ferent values of δ. All other parameters were kept constant. The distributions have been normalised by their respective maximum values. 32 Chapter 5. Numerical simulations and results rate and thus most captured particles result from collisions with very similar small recoils. The thing that determines where they end up is then the energy they had before the collision. The flatter part below the capture maximum would then correspond to the steep rise in the velocity distribution before the peak, and these particles are captured to the lowest energies. Then we get the capture maximum around the peak of the velocity distribution where most of the particles reside, and above the maximum we see the more extended tail of the distribution with particles of higher energy. The fact that the capture rate distribution is wider above the capture maximum than below it for δ ≤ 100 keV is attributed to capture on different target elements. As we increase δ above this value the contribution due to any elements but iron become negligible. That the above is the case should be straight forward to verify by changing to another DM halo velocity distribution and rerunning the simulations, something which is not done here, however. In the bottom right panel of Figure 5.2 the capture rate is shown in the large 10 mass limit of the mediator, here it is taken to be mB0 = 10 keV. This reproduces well the left panel in Fig 2 in Ref. [11]. It should also be noted that the computed total capture rate coincides with that in Ref. [50] to within 0.2 % in this limit with the use of the Helm form factor. In this thesis we are mainly interested in the shape of the capture rate distri- bution and thus not too much focus will be placed on the absolute values of the capture rate. The capture rate itself depends heavily on our choice of parameters and if we were to select another value for σR or qR the total capture rate would change. Despite this it is still presented in the figures to show how a change of parameters influences the total capture rate. While the mass of the mediating particle is constrained by δ, no real difference in either shape or total capture rate is expected. This is confirmed by comparing the computed capture rate for mA0 = 20 keV and mA0 = 40 keV for all values of δ. One example, with δ = 100 keV, of the capture rate for these two different mediator masses is shown in Figure 5.3. The momentum transfer in the involved scattering events is far greater than any mA0 < δ and completely outweighs any such differences. It is only when the mediator mass is large enough to actually compete with q that one would expect the result to depend on changes in this parameter. Considering a scenario where different particles are involved in the decay of DM and in scattering with target nuclei we observe what happens for larger masses of the scattering mediator particle. The result is shown in Figure 5.2. A rather nice transition can be observed to take place: As the mass mB0 of the scattering mediator is increased to values comparable to, or greater than, the typical involved momentum transfer the distribution gains a broader shape and starts to grow. As the suppression for momentum transfers other than the minimum possible are not as severe in these cases, a wider variety will be allowed leading to the larger region. Notice, however, that a larger capture region does not equal a larger total capture rate. Up until mB0 = 50 MeV the total capture rate can be seen to decrease, while after that it starts to increase again. This behaviour is due to our parametrisation of the cross section in eq. (3.47) and choice of qR. 5.2. Solar capture rate 33
Figure 5.2: The total capture rate into the different E-L states for different values of the scattering mediator particle mass mB0 . All other parameters were kept constant. The distributions have been normalised by their respective maximum values. 34 Chapter 5. Numerical simulations and results
Figure 5.3: The computed total capture rate into the different E-L states for the different mediator masses mA0 = 20 keV (left) and mA0 = 40 keV (right). The distributions have been normalised by their respective maximum values.
Our choice of qR is on the lower side of involved momentum transfers and as we increase the mediator mass the suppression impact of allowing momentum transfers slightly above the minimal one is reduced. This allows for slightly larger momentum transfers to contribute which lower the overall capture rate. As the mediator mass becomes larger and larger, the involved momentum transfer lose influence and the expression in eq. (3.47) start to tend towards σR.
5.3 Solar scattering rate
In Figure 5.4 the scattering rate for each state multiplied by the lifetime of the Sun is shown for different values of δ. In general we see that less scattering takes place for larger values of δ and that the peak of the scattering rate moves towards less bound states and lower angular momenta for larger δ. The transition in energy is due to the fact that a larger kinetic energy is needed to scatter as δ is increased. The fact that we move away from circular orbits, i.e., orbits with maximal angular momentum, towards those that are more elongated can be understood in a similar manner. The elongated orbits are the ones that give the DM particles opportunity to accelerate and gain more velocity when moving from larger radii to smaller ones, thus providing them with the required velocity to scatter. These orbits will also take the particles closer to the centre of the Sun, where heavier target particles are encountered and the density greater. These are the primary contributors to the behaviour of the scattering rate, increasingly so as δ becomes larger. We can also take the limit where the mediator becomes very heavy, as we did for the capture rate, in order to recover earlier results. We of course have to disregard any decay in this case, or consider the scenario with two different particles. When 5.3. Solar scattering rate 35
Figure 5.4: The base 10 logarithm of the total scattering rate in each E-L state multiplied by the current age of the Sun for different values of δ. ignoring any contributions due to the decay of the heavier state and taking the large mass limit of the mediator we obtain the result shown in Figure 5.5. This figure can be compared with the left panel of Figure 3 in Ref. [11]. They are again strikingly similar, as is expected if everything behaves as it should. Comparing the top right panel of Figure 5.4 with Figure 5.5 we can observe the effects that the light mediator have on scattering. The structure is in general very similar, but we do have a much smaller region where the highest scattering rates are found. This is due to the additional suppression in the cross section introduced by the mediator particle, narrowing down the high scattering region only to those events that have minimal recoil energy. In Figure 5.6 we again consider the case of a second particle, separate from the decay process, that mediate the scattering process and we gradually increase its mass and observe the results. As we move towards larger mediator masses the region of large total scattering again starts to grow, confirming the conclusions above. 36 Chapter 5. Numerical simulations and results
Figure 5.5: The base 10 logarithm of the total scattering rate multiplied by the 10 current age of the Sun for scattering mediator particle mass of mB0 = 10 keV.
5.4 Time evolved distributions
We now use our computed capture and scattering rates to look at how the DM distribution in the Sun evolves with time. The initial distribution is taken as the capture rate vector normalised by the total capture rate, i.e.,
C f(0) = P . (5.12) α Cα This choice is in the end irrelevant as it will cancel out when we normalise the evolved distribution. We evolve the initial distribution in time using the scattering matrix, and the furthest we evolve it to is t = 10t . The result is shown in Figure 5.7. We obtain two main regions where particles seem to accumulate, one at the very bottom left and one at the top right. The group at the bottom left has several orders of magnitude more particles and this is where particles tend to move to from their initial states. The area in the top right consist simply of those particles that were captured into these states to begin with, these have not scattered at all. This can be seen to be the case by looking at the top right panel of Figure 5.4 and noticing the area with zero scattering rates in the top right. Observing the distributions for larger values of δ confirms this to be the case as the area grows precisely in the same manner as the zone where no scattering occur. We can further verify this to be case, and strengthen the case for all particles to move towards the lower left, by looking at where particles tend to go when scattering. This information is stored in the scattering matrix, and an extra benefit is that this may also provide information on whether evaporation occurs or not. Carrying this out, we notice that no state seems to scatter to states that have higher energy, scattering always results in the particle having less energy than it had before. An example of this is shown in Figure 5.8. As particles need to lose 5.4. Time evolved distributions 37
Figure 5.6: The base 10 logarithm of the total scattering rate multiplied by the current age of the Sun for different values of the scattering mediator particle mass mB0 . 38 Chapter 5. Numerical simulations and results
Figure 5.7: The base 10 logarithm of the DM particle number distribution at differ- ent stages of the time evolution process. The distribution is normalised such that RR f(E,L) dE dL = 1 at t = 0. 5.4. Time evolved distributions 39
Figure 5.8: Examples of where particles in a specific state α, marked in red, scatter to. The left column shows the base 10 logarithm of the total scattering rate multi- plied by the current age of the Sun for all E-L states. The right panels show the base 10 logarithm of the rate at which a specific state α, marked in red, scatters to all other states β 6= α multiplied by the current age of the Sun. a substantial amount of energy simply to accommodate scattering, this is not too surprising. Observing the progression in Figure 5.7, it is seen that all major changes oc- cur within the Sun’s current lifetime. Subtle changes can however still be seen when moving further in time, indicating that a complete steady state has not yet been reached. The exact time scales for this progression is of course dependent on our choice of parameters in eq. (3.48) which, as a brief reminder, we took as −42 2 σR = 10 cm and qR = 50 MeV. The region that traces the right edge of the dis- tributions in between the two unchanging areas is rather interesting since it changes so very little during such a large time-span. We noted earlier that the scattering rates for the states with high angular momenta are in general much lower than the rest. Adding to this we can note that they generally tend to scatter to other states 40 Chapter 5. Numerical simulations and results
Figure 5.9: Top left panel shows the base 10 logarithm of the DM number distri- bution at the current age of the Sun. The remaining three panels show the base 10 logarithm of the scattering rate at which a particle in a specific state α, marked in red, scatter to all other states, β, multiplied by the current age of the Sun. of high angular momentum while moving down in energy. In Figure 5.9 we trace where particles along the right edge of the distribution tend to go. Particles right on the edge of the group to the upper right slowly, but surely, make their way down towards the majority of particles in the lower left. We now transform the E-L distribution and obtain the radial distribution. The radial distribution corresponding to the E-L distribution shown in Figure 5.7 is shown in Figure 5.10. In these figures we also show the isothermal Boltzmann distribution given by eq. (4.21) using the core temperature and core density of the Sun taken from AGSS09ph [69]. If the DM particles were to reach equilibrium far down in the core of the Sun we would expect the DM distribution to resemble the isothermal one. A first observation is that this is clearly not the case, even for t = 10t . We again note that some very minor changes do occur as we move from t = t to t = 10t . 5.4. Time evolved distributions 41
Figure 5.10: The radial distribution of DM at different stages of the time evolution process. There are in total 100 bins, each corresponding to 1 % of the solar ra- dius. An isothermal Boltzmann distribution corresponding to the temperature and density of the solar core is also shown. 42 Chapter 5. Numerical simulations and results
Figure 5.11: The radial distribution of DM for different values of δ at t = 10t . An isothermal Boltzmann distribution corresponding to the temperature and density of the solar core is also shown.
The resemblance does grow stronger as we move down in δ, as shown in Fig- ure 5.11. This is in agreement with the fact that elastic DM thermalises rather quickly, and does in fact follow a Boltzmann distribution rather well [11, 68]. The radial distribution computed in the heavy mediator mass limit is shown in Figure 5.12 and the left panel should be compared to the rightmost panel in Figure. 7 in Ref. [11]. They can be seen to very similar both in height and width. One interesting thing to note is that the presence of the light mediator seems to have only a very small impact on the end result for the radial distribution, the top right panel of Figure 5.11 and the right panel of Figure 5.12 are extremely similar. This fact may not be that much of a surprise. Particles will of course tend to move away from the regions where the scattering rate is largest rather quickly, as we saw in Figure 5.7. Even though the regions with largest scattering rates differ a bit with and without the mediator, the behaviour of the scattering events is the same. All scattered particles move down in energy towards the regions where the scattering 5.4. Time evolved distributions 43
Figure 5.12: The radial distribution in the heavy scattering mediator limit for the current age of the Sun (left panel) and ten times the current age of the Sun (right panel). An isothermal Boltzmann distribution corresponding to the temperature and density of the solar core is also shown. matrices look the same: The “no scattering zones” in the top right and bottom left are almost identical and the regions where log10(Γ(E,L) · t ) < 9 are also in general very similar. Thus as we move past the initial stages in time evolution they should behave in a very similar way.
Chapter 6
Summary and conclusions
In this thesis we have investigated the thermalisation of inelastic DM in the Sun in the presence of light mediator particles. The relevant kinematic relations were computed and presented and thereafter numerical simulations of the process were performed. The light mediator mainly introduces new complications in the scat- tering cross section, where it leads to additional momentum transfer dependencies. This extra dependence efficiently suppresses a lot of the recoils that are otherwise allowed, especially when the mass of the mediating particle is relatively small. This leads to a smaller region where capture can occur and information about the veloc- ity distribution of DM before capture seems to have a noticeable effect in shaping this region. The scattering of captured particles is not changed as significantly and the observed change is that the regions with highest total scattering is smaller with the mediator than without it. The effects of the mediator on both capture and scattering decrease as we increase the mediator mass, as expected, and in the limit where the mass is very large we recover earlier results from Refs. [11, 50]. No steady state seems to be reached and it is clear from the radial distributions that we are far from thermal equilibrium, as described by an isothermal Boltzmann distribution. We do note, however, that as we decrease the mass difference between our two DM states, the resemblance seems to improve, this is also expected from earlier studies. In general, the radial distributions found are extremely similar to both the one obtained in the high mediator mass limit and to those obtained in studies without mediators. There are no clear distinct features for the radial dis- tributions, as opposed to both the capture rate and total scattering rate, where noticeable differences could be seen. It seems that most of the information con- cerning the initial captured distribution is lost rather quickly and past the initial stages of time-evolution the scattering processes with and without the mediator are very similar. There are several paths one could take to extend this work. First of all, simula- tions of elastic DM involving a light mediator would be a natural place to start. At- tempts at this were made during the project, however numerical issues not present
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