WISPy Dark Matter Axion-like particles as dark matter candidates

(1)

WISPy Dark Matter

Axion-like particles as dark matter candidates

Lino Kragtwijk

Supervisors:

Dr. Christoph Weniger

Dr. Shin’ichiro Ando

Bachelor Project Physics and Astronomy

15 EC

GRAPPA

University of Amsterdam

Netherlands

(2)

1 Summary

Axion-like particles look promising as dark matter candidates. Their production by the misalignment mechanism ensures a sufficient amount of particles in the early universe allowing structure formation. The characteristics of the axion have been investigated, starting with the decay channel in two photons. A detailed description of the process is provided in appendix A. The density distribution of the dark matter halo in the Draco galaxy is modeled. Using line of sight integrals over this density, we calculated the flux. By using an interpolation technique the background has also been determined, leading to the use of the Fisher information for significant results. The constraints on axions as dark matter have been sharpened as shown by the graphs.

2 Populair Nederlandse samenvatting

Donkere materie, een onderwerp dat nogal tot de verbeelding spreekt, maar wat is het eigen-lijk? Een groot deel van ons universum, maar liefst 23% van alles zou uit donkere materie bestaan. Het is ook heel belangrijk, zonder donkere materie zouden er namelijk geen sterren-stelsels zijn onstaan zoals wij ze kennen. Wat het zo moeilijk maakt om te bepalen wat het is, is dat het nauwelijks interactie heeft met materie die we kennen. Het enige dat we zien is dat er meer massa is dan de theorie voorspeld. In dit project heb ik gekeken of een bepaald deeltje, het axion, het zou kunnen zijn. Daarvoor heb ik eerst onderzoek gedaan naar het axion, wat zijn de eigenschappen, hoe vervalt het? De motivatie om het axion is te kiezen is dat het in grote getallen geproduceerd kan worden in het vroege universum en een lange levensduur heeft. Dit zorgt ervoor dat er aan een aantal voorwaarden worden voldaan voor donkere materie, zoals het mogelijk maken van structuren in het heelal. Wat fijn is aan het axion, is dat door zijn kleine massa het aantal vervalproducten beperkt is tot voornamelijk fotonen. Tijdens het onderzoek heb ik gekeken naar de verdeling van donkere materie in het Draco stelsel. Met integralen over de dichtheid langs de lijn waar je mee kijkt, heb ik een flux bepaald. Dit heb ik samen met de achtergrond straling van het universum bekeken. Zo is bepaald aan welke voorwaarden het axion zou moeten voldoen om het met bepaalde zekerheid waargenomen te worden. In de grafieken zijn de voorwaarden voor het axion als donkere materie aangescherpt.

(4)

3 Introduction

Dark matter has a suitable name as it is a subject that is veiled in mystery. The effects of it are clear to see, but what is dark matter? At first it was an easy solution to explain trajectories and their orbital velocities in the universe. The velocities did not align with current theories, which led to the hypothesis that there was more mass than was detected (Zwicky 1933). This lack of mass was explained with dark matter, a hypothetical kind of matter that couldn’t be detected at the moment. In the present day, the ideas about what it could consist of are more refined. A lot of restrictions have been added to the kind of particle dark matter could consist of. In this project I have researched the possibility of axion-like particles (ALP) being the elusive dark matter particle. There are reasons to believe axions are the dark matter particle, which will be discussed in detail. The project can be divided in sections, where each section provided a step towards the conclusion. The first step was to define the axion. The axion is confined in the interactions and the possible masses it can have. The axion decays in two photons, this calculation is reproduced to find the decayrate and the lifetime of the axion. The next step was to model the dark matter density in the Draco galaxy. The Navarro Frenk White model has been used for this. The modeling is done in Python, from the density a flux can be calculated by using line of sight integrals. The final step was to check the background. The background from the universe was modeled using recent papers (Cooray 2016). We conclude by suspecting whether the SPICA telescope could detect the axion decay. To draw a good conclusion, the results were calculated while using the Fisher information. This theory was used to calculate the time needed to detect the particle with a certain significance. To explain the title, WISPy stands for Weakly Interacting Slim Particles which the ALP fullfills the conditions for. Improvements on the calculations can be made by taking more details into account, more on this in the discussion.

4 Dark matter

In this section a more thorough research of dark matter will be described. The history and the physics which led to the conclusion there should be dark matter will be discussed first, after that the conditions which dark matter must obey will be discussed next. This will give rise to the cold, hot and warm types of dark matter in relation to structure formation in the universe. Theories of particle candidates will be discussed.

(5)

4.1 History of dark matter

Dark matter was first suggested by F. Zwicky who studied the Coma cluster in the 1930’s (Zwicky 1933). A galaxy cluster is, as the name suggests, a group of galaxies. The calculations for radial velocities on these large structures have some peculiarities since the objects don’t orbit a massive object, which makes their trajectories more complicated. Using a telescope, Zwicky measured the spectra of the Coma cluster. By calculating the Doppler shift of the spectra he was able to deduce velocities. He then computed the mass using the virial theorem. This theorem relates the kinetic energy averaged over time to force and position of objects under potential forces:

hT i = −1 2 N X k=1 hF_k· r_ki (1)

Where Fk is the force on the kth particle and rk is the position of that object. For gravity

the equation can be rewritten as:

hT i = −1

2hV i (2)

With use of these expressions, Zwicky concluded that the radial velocities were higher than the mass to light ratio would suggest. Objects in orbit due to gravity are expected to slow down the more outwards the object is. However, this was not what was detected. The mass of the cluster which was detectable with light was not enough to explain the velocities. When compared with other data from a stellar system, Zwicky concluded there maybe was ”dark (cold) matter” to increase the density in the galaxy.

At first, dark matter didn’t have much attention in the scientific community. The universe was large and complex, general relativity could be tested around the same time and other phenomena emerged. With more precise measurements, dark matter became more interesting. In 1973, Ostriker and Peebles made a numerical analysis of galaxies (Ostriker & Peebles 1973). Their method was to simulate the mass points orbitting a central point using a N-body simulation. Interestingly, their conclusion was that to get the current galaxies, a dark matter halo distribution had to be added. At the same time, Ford and Rubin found empirical evidence for dark matter in their study of the Andromeda nebula (Rubin & Ford Jr 1970). Further data lead to their paper in 1980, which convinced astronomers that dark matter was needed to explain rotational curves. Figures 1 and 2 represent the data and the evidence which stems from it.

(6)

Figure 1: Data published by (Rubin et al. 1980).

Figure 2: Measured and projected velocities which point to dark matter, B is the curve with dark matter, looking like the curves found by (Rubin et al. 1980).

4.2 Conditions for dark matter

Dark matter obeys certain conditions. These conditions are based on empirical evidence on one part and theoretical restrictions on the other part. The empirical evidence part is fairly

(7)

intuitive, we only see dark matter as an indirect effect. As such it is assumed that dark matter has little interaction with ordinary matter, other than gravitational presence. The range of objects which can fill this spot is huge, from brown dwarves to neutrinos. Further restrictions downsize the pool of candidates. Production of the objects, cosmological stability and other factors are examples of such restrictions. The candidates need to have a long lifetime, be produced in abundance to ensure enough mass and be non dissipative enough to allow structure formation. There are three distinct classes of candidate particles based on their dissipation in the universe.

4.3 Types of dark matter

There are three distinctions in dark matter called cold, hot and warm dark matter. The cold, hot and warm terminology doesn’t refer to temperature but rather to the speed of the particles. Hot dark matter (HDM) consists of particles moving at ultrarelativistic speed. Candidate particles would be neutrinos, however theory excludes hot dark matter as the sole constituent of dark matter (Frenk & White 2012). HDM simulations don’t result in the current universe. It is important to note that the three classes have different smallest structures. In HDM superclusters form first and these must fragment to make smaller structures. Warm and cold dark matter start with small objects which merge and accrete to bigger systems. This has to do with the freestreaming length of the particles. The freestreaming length (FSL) is the distance a particle can travel before becoming non relativistic. Since heavier particles are assumed to have a smaller FSL, the densities at which they clump together are small in comparison to faster, lighter objects.

Cold dark matter (CDM) consists of particles who move relatively slow in the early expanding universe. Weakly interacting massive particles (WIMPs) and axions could be the constituents of CDM. WIMPs were suggested by Peebles (1982) and are currently the most researched candidates. They haven’t been found yet, research at the Large Hadron Collider in Geneva hasn’t given the final call. Warm dark matter is a hypothetical particle which is between CDM and HDM. It allows galaxies to form, starting with structures smaller than what HDM produces but bigger than what CDM produces.

5 Axions

An important characteristic of dark matter is their weak interaction with ordinary matter. This restricts the choice of particles which could be dark matter. A suitable candidate comes forth from Quantum Chromodynamics (QCD). In QCD, there is a problem with charge parity (CP) operations. Where in other areas of physics these operations are violated, in QCD they

(8)

aren’t. By violated it’s meant that changing charge and parity doesn’t change matter to anti-matter and vice versa. This means that matter and anti-matter are different and can’t be interchanged by operations. This was first looked at by Landau (1957) for neutrinos. He found discrepancy in the weak interaction, resulting to the discovery of CP violation.

5.1 Peccei-Quinn solution

A solution to the CP problem was proposed in 1977 by Peccei & Quinn (1977). An effect of their solution is the production of pseudo Nambu-Goldstone-bosons which are called ax-ions. The axions are produced by the misalignment mechanism in the early universe. The misalignment mechanism occurs when an initial field value is not at the potential minimum. This causes ossicilations in the field, which are essentially particles. This is shown in figure 3.

Figure 3: Representation of the misalignment mechanism with a Mexican hat potential

This mechanism could produce enough axions to meet the requirements for dark matter (Preskill et al. 1983). For a more detailed overview of the mechanism I refer to (Arias et al. 2012), which was the lead paper in my project.

5.2 Characteristics

The axion in QCD has restrictions on the mass and lifetimes. In this project axion-like parti-cles are considered, free of some restrictions. The mass of the axion is in the low eV,therefore the possible decay channels are dominated by photon decay. The lifetime of axions is suffi-cient for structure formation. Interactions of axions with matter are explained with quantum field theory (QFT). The coupling of the axion with the electromagnetic field is given by the Lagrangian density in (3):

(9)

L = −1

4gφFµνF˜

µν ₍₃₎

Here g is the coupling constant, φ the field, and F stands for the electromagnetic interaction term. From this equation one can get the coupling matrix M, which is needed in the decay calculation. For a detailed derivation there is appendix A.

5.3 The decay

The decay of the axion is represented by the Feynman diagram in figure 4.

Figure 4: Representation of the decay of the axion in 2 photons

To find the lifetime of the axion it’s needed to evaluate the diagram. In this project, there is no loop and the interaction happens in a single vertex. General equations for decay can be found in (Thomson, 2013). The following equation is used:

Γ = 1 2mφ Z d3p1 (2π)3 d3p2 (2π)3 1 E1 1 E2 | M |2 (2π)4δ4(pφ− p1γ− p2γ) (4)

This is the equation corresponding to figure 4, for calculation of the decay. The M has to be calculated from the Lagrangian, this is shown in appendix A. When putting in the parameters and evaluating the integrals, the equation reduces to (5) (Arias et al. 2012). Deriving the result is a tedious process and is discussed in detail in appendix A.

Γ = g

2_m3 φ

64π (5)

6 The research method

The signal of the dark matter will be researched with the SPICA telescope. The SPICA telescope will be launched in the future,therefore the research has been conducted on

(10)

theoret-ical grounds. The specifications of the satelite allow finetuning of the researched areas. The values of importance have been set out in a table:

Table 1: Table with the needed specifications.

SPICA Specifications

Detection range 5 to 210µm or 0.00496 to 0.248 eV Lifetime SPICA is operational for 3 to 5 years Accuracy Up to 10−3 of the signal energy Solid angle resolution 2’x 2’

Area r = 1.75m so the area is 9.62m2

6.1 The dark matter signal

The decay process has been explained, now the expected signal will be discussed. The Draco dwarf galaxy has been chosen as the object of observation. The Draco system is well known for its dark matter properties (Tyler 2002). The mass to light ratio M_L was found to be 440 ± 220 M

L, while the mass M is approximately 8.6 ∗ 10

7_M

. M and L are the mass and

luminosity of the Sun. These numbers indicate a dark matter dominated galaxy. The mass is isothermally distributed, with a steep central density. The dark matter halo of the Draco galaxy is distributed according to the Navarro-Frenk-White profile:

ρ(r) = _r ρ0

rs(1 +

r rs)

2 (6)

ρ0 and rs are constants which vary from halo to halo. The scale radius rs is a cutoff value

for describing a r−1 or r−3 function for that halo. The density ρ0 is the density for the halo.

They are calculated by solving the equations for the concentration-mass relation with the following equations (Huang et al. 2011):

cvir(Mvir) = 9 Mvir 1014_h−1_M −0.172 (7) c = rvir rs (8)

(11)

f rs r∆ = ∆ ∆vir f rs rvir (9) where f (x) = x3[ln(1 + x−1) − (1 + x)−1] (10) and M∆ Mvir = ∆ ∆vir r∆ rvir 3 (11) M∆ and r∆ are the cluster mass and radius, which can be observed. These equations relate

the cluster mass and radius to the scale radius and density parameters.

By using line of sight integrals over the density, one gets the signal from one pixel of the telescope.

The geometry of line of sight integrals in illustrated in figure 5. There’s a change of variables that has to be done, since the density is described from the centre of Draco and our coordinate system starts from SPICA. This solved by drawing a line with a 90 degree angle from the centre to the line of sight. The problem then simplifies to calculating the radius with a change of variables.

Figure 5: Representation of a line of sight.

The signal expected for the axion decay is calculated using a Python script. The expected flux is described by an integral over all line of sight integrals that cover the sphere:

dJsignal dE = 1 mφ 1 4π 1 τφ Z ∆Ω Z l.o.s. ρdrdΩ (12)

(12)

The units of the NFW profile were converted from M

M pc3 to eV

cm3, the final output is given in

units of photons_cm2_s . From the SPICA information we concluded to evaluate only up to an angle

of 1’ in a circle. This translates to a value of 2.658288 × 10−7 steradian.

6.2 Background signal

In order to conclude anything about the detection, the background light of the universe has to be evaluated. A recent publication (Cooray 2016) gives insight in the background, by combinating multiple publications. The results are shown in figures 6 and 7.

Figure 6: The whole spectrum of the extragalactic background .

The background signal is constructed by taking values of the graphs and using an interpolation routine to create a function. The units of the graphs have to be converted to fit our previous calculations for the flux. This will be explained in the next part.

6.3 Fisher information

To determine whether the results are statistically significant, the Fisher information equation is used: I = T A Z _φ signal2 φbackground dE (13)

The F.I. scales with the square of the standard deviation (Weniger 2016). Here T is the time for detection and A the area of detection. The time of measurement has varied from 100 seconds to a year, the area of the satelite is given by table 1. The function for the signal is:

(13)

Figure 7: A zoomed in version of the spectrum relevant to the project . φsignal= dJ dE 1 σ√2πe (E− 1₂_maxion)2 2σ2 (14)

The φsignal contains signal flux for the dark matter, spread with a Gaussian distribution for

a fixed mass. The parameters for the Gaussian distribution are given by the decay and the SPICA restrictions. The σ comes from table (1), the most probable value is half of the mass of the axion. The boundaries for the integral are given by the detection range of SPICA (0.00496 - 0.248 eV). The units of φsignal are E−1.

The signal for background given by νIν is in units of nW m−2 sr−1. This had to be converted:

νIν = E

dI dlogE = E

2 dI

dE (15)

When taking a point from figure 7, the wavelength was translated to energy with: E = hc

λ× 6.24110

18_eV ₍₁₆₎

Going from nW m−2 sr−1 to eV cm−2sr−1 adds a factor of 6.241 × 1014 to each data point. By dividing the intensity by the energy of each datapoint twice, the correct units and signal for the background are found. This is then multiplied by the amount of steradial derived in section 6.1 (2.658288 · 10−7 sr). The data points are then converted to a function by using interpolation between the points.

(14)

The next step in the research was to choose a value for the Fisher information. It has been set to a value of 4, corresponding with 2σ. By making a grid of values, figure 9 was produced. First the Fisher function was rewritten as a function of lifetime and mass, set on a constant significance. Next, each iteration calculated the needed lifetime given a certain mass to fit the 2σ chosen for the F.I. The masses and lifetimes of the axion corresponding to this method are plotted on figure 9.

7 Results

The first plot is used to gain insight in the flux for different coupling constants, shown in figure 8. The other plot shows the plots of mass-lifetime relationship for a 2σ significance for different detection times. Previous restrictions have been added to the graph, the original comes from Arias et al. (2012) in figure 10.

Figure 8: Flux of the axion for different masses and coupling constants .

(15)

Figure 9: The mass-lifetime relationship is computed for different detection times at a 2 σ significance

.

Figure 10: The restrictions on mass and lifetime as known for current research (Arias et al. 2012)

(16)

8 Discussion

Figure 8 is intuitive, it shows what the flux is for different values. The second graph shows the restrictions, for an atleast 2σ significance. The lines indicate the allowed lifetimes for a given observation time, still fulfilling the Fisher information requirement. To make clear which way the limit is restricted, going lower in the lifetime direction makes the flux higher. A signal was already expected there, therefore the excluded area is above the datapoints.

The data is not perfect, a lot of assumptions have been made in the process. The models can be improved by more precise data, improving the expected signal with more refined integrals, taking ellipcity into account for Draco and refining the background signal.

9 Conclusions

The plots show that constraints on dark matter have been improved under the conditions used in this project. Given enough observation time, SPICA might be able to detect axions, finding the dark matter particle. The results point out that there is room for improvement on the topic of axions as dark matter particles, given the level of research conducted. The line of sight integrals conducted do however provide a basis. Following with the background signal in combination with the Fisher information gives a foundation to the project. This project had the intention of exploring axions as dark matter particles. The process has shown progress in the field of dark matter. A little light has been shed on the dark.

(17)

10 Acknowledgements

I would like to thank Christoph and Thomas for their guidance throughout the project. They took time to make sure I understood the subjects, more than just being able to compute stuff. The project has given me a good insight how the master phase of my studies will look like, I look forward to this new challenge. I’m happy to have conducted this research and respect the effort people before me have made in this challenging field.

References

Arias, P., Cadamuro, D., Goodsell, M., et al. 2012

Cooray, A. 2016, Extragalactic Background Light: Measurements and Applications Frenk, C. S. & White, S. D. M. 2012, Dark matter and cosmic structure

Huang, X., Vertongen, G., & Weniger, C. 2011 Landau, L. 1957, Nuclear Physics, 3, 127

Ostriker, J. P. & Peebles, P. J. 1973, The Astrophysical Journal, 186, 467 Peccei, R. D. & Quinn, H. R. 1977, Physical Review Letters, 38, 1440 Peebles, P. 1982, Astrophys. J, 263, L1

Preskill, J., Wise, M. B., & Wilczek, F. 1983, Physics Letters B, 120, 127 Rubin, V. C. & Ford Jr, W. K. 1970, The Astrophysical Journal, 159, 379

Rubin, V. C., Ford Jr, W. K., & Thonnard, N. 1980, The Astrophysical Journal, 238, 471 Tyler, C. 2002, Physical Review D, 66, 023509

Weniger, C. 2016, Personal conversation about a to be published article Zwicky, F. 1933, Helv. Phys. Acta, 6, 110

(18)

11 Appendix A

11.1 Calculating the decay

The detailed decay calculation is explained here. We start off with the Lagrangian.

L = −1 4gφFµν ˜ Fµν = −1 8gφµνρσ[F µν_Fρσ_] ₍₁₇₎ FµνFρσ = (∂µAν − ∂νAµ)(∂ρAσ− ∂σAρ) (18) = ∂µAν∂ρAσ− ∂µAν∂σAρ− ∂νAµ∂ρAσ+ ∂νAµ∂σAρ (19) By changing order in the Levi-Civita symbol and changing variable we get:

µνρσFµνFρσ = 4µνρσ∂µAν∂ρAσ (20)

By using the identities:

∂µAν = −ikµAν (21)

rs= δrs (22)

We can write the M as this:

M = gνσ_kµ_k0ρ ₍₂₃₎

Squaring the matrix and summing over the spins gives the following results:

spins

X

| M2 |= g2αβµραβγδkµkγk0ρk0δ (24)

= −2g2(gµγgρδ− gµ∂gργ)kµkγk0ρk0δ (25)

= −2g2(k2k02− (k · k0)2) (26) Filling this into equation (4) for the decay, we get:

Γ = 1 2mφ Z _d3_p 1 (2π)3 d3p2 (2π)3 1 E1 1 E2 | M |2 (2π)4δ4(pφ− p1γ− p2γ) (27) Γ = pf inal 32π2_m φ2 Z | M |2 _dΩ ₍₂₈₎

(19)

Where the k · k0 in the | M |2 is equal to 2E2 , kk0 is 0. The energy of the photons is always half the mass of the axion in the c.m. frame. Filling this in gives:

Γ = g 2 64π2_m φ 4mφ 2 4 × 2π = g 2_m3 φ 32π (29)

Because we are dealing with identical particles, we have to divide by 2, giving the result: Γ = g

2_m3 φ

WISPy Dark Matter Axion-like particles as dark matter candidates