There is a crushing evidence from the “Bullet Cluster”(Figure 2.2), which is a result of a smaller subcluster colliding with another larger one. During the merger of two clusters, the galaxies inside both clusters behave as collisionless particles passing through each other without interacting, while the hot gas(the majority of baryonic matter) experiences ram pressure. After the collision of clusters, a huge amount of X-ray radiation can be observed because of the compressed and heated gas. If we match the location of the radiation with the location of the majority of the total mass of the clusters, which is measured via weak gravitational lensing, the result shows a discrepancy. We conclude that the majority of the mass is non-barynoic.
7
Chapter 3
PARTICLE CANDIDATES IN SM?
According to previous introductions, though there has no clear definition of a WIMP, it is possible to seek if there exists a candidate in a theory due to the gravitational effect and other properties, say,it doesn’t interact with light.
The Standard Model is the quantum field theory that describe three of the four funda-mental forces in the Universe: the electromagnetic, weak, and strong interactions, also clas-sifying seventeen elementary particles which were confirmed experimentally(see Table 3.1).
There are six quarks(up ,down, charm, strange, top, and bottom), six leptons(electron, e neutrino, muon, µ neutrino, tau, and τ neutrino), and five bosons(photon, W±, Z, gluon, and Higgs). Quarks and leptons are classified as fermions with half-integer spins, where bosons with integer spins. Every particle has its corresponding anti-particle.
In SM, the only stable, neutral, and weakly interacting particles are neutrinos. Neutri-nos are not the candidate because they are relativistic, that is to say, if the Universe was dominated by neutrinos, then the formation of the Universe would have started from larger structures; however, the “bottom-up” formation seems to be the most possible case since galaxies have been observed to exist less than a billion years after the big bang. Thus, new physics is required to supply the Standard Model.
8
Fermions
I II III
Particle Mass Charge Particle Mass Charge Particle Mass Charge
u ≈ 2.2 +23 c ≈ 1280 +23 t ≈ 173000 +23
d ≈ 4.7 −13 s ≈ 96 −13 b ≈ 4180 −13
e− ≈ 0.511 −1 µ− ≈ 106 −1 τ− ≈ 1777 −1
νe − 0 νµ − 0 ντ − 0
Bosons
Particle Participate in Mass Charge
γ Electromagnetic interaction 0 0
W±
Weak interaction ≈ 80385 ±1
Z ≈ 91188 0
H0 Higgs field ≈ 12509 0
Table 3.1: The particles in the Standard Model. The masses are in units of MeV reported by Particle Data Group[5].
9
Chapter 4
INDIRECT DETECTION
The products of annihilations or decay of DM can be the SM particles, hence they will become new primary sources of cosmic ray particles; and further changing the spectra of anti-particles such as positron and anti-proton, while they are assumed to be secondaries in the conventional cosmic ray model. Detecting the abundance and the flux of these species offers a good chance to examine the properties of DM. The term “indirect” is because the experiments are detecting the final states of DM(i.e. the SM particles) instead of WIMPs themselves.
The Alpha Magnetic Spectrometer (AMS-02) is a module which is mounted on the International Space Station(ISS) for the purpose of measuring anti-matters (anti-particles) in cosmic rays. It is the most sensitive particle detector so far in space. As a magnetic spectrometer, it has the ability to distinguish charged particles. AMS has collected over 90 billion cosmic ray events since its installation on ISS. Protons are the most abundant particles, which has been measured the flux to an accuracy of 1% with 300 million protons and found that the flux deviates from a single power law, as had been assumed for many years[6].
The boron (secondary cosmic ray) to carbon (primary cosmic ray) flux ratio measured by AMS provides significant information on propagation and the average amount of interstellar
10
material (ISM) through which the cosmic rays travel in the galaxy. Cosmic ray propagation is commonly described as a relativistic gas diffusing through a magnetized plasma. Different models of the magnetized plasma predict different behavior of the B/C flux ratio. Notably, the B/C flux ratio data from AMS does not show any crucial structures in contrast to many cosmic ray models which don’t include DM.
In 2016, the AMS collaboration released the latest result of anti-proton to proton flux ratio (¯p/p). It shows that the flux ratio stays constant or even a little excess from 20 to 450 GeV. This cannot be explained by the secondary anti-proton, which is from the collision of cosmic rays with ISM. In this work, we will discuss the contribution from DM by calculating its propagation and fitting the AMS data.
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Chapter 5
NUMERICAL RESULTS
The propagations are calculated using GALPROP[7, 8, 9, 10, 11]. It solves the transport equation with a given source distribution and boundary conditions for all cosmic-ray species.
The equation is written in :
∂ψ
∂t = q (~r, p) + ~∇ · (Dxx∇ψ − ~~ V ψ) + ∂
∂pp2Dpp ∂
∂p 1 p2ψ
− ∂
∂p[ ˙pψ − p
3( ~∇ · ~V )ψ] − 1 τf
ψ − 1 τr
ψ, (5.1)
where ψ = ψ(~r, p, t) is the number density per unit of total particle momentum, q(~r, p) is the source term, Dxx is the spatial diffusion coefficient taken as Dxx = βD0(ρ/ρ0)δ, where ρ = p/Ze is the rigidity of the cosmic ray particle with electric charge Ze, and β(= v/c) is a consequence of a random-walk process. The power index δ can have different values δ = δ1/2 when ρ is below/above rigidity ρ0. The convection velocity ~V is related to drift of cosmic-ray particles from the galactic disc due to the galactic wind, which implies a constant adiabatic energy loss. The re-acceleration is described as diffusion in momentum space and is determined by the coefficient Dpp. The momentum loss rate ˙p(= dp/dt) could due to ionization, Coulomb interactions, bremsstrahlung, inverse Compton scattering, and
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Model R(kpc) Zh(kpc) D0 ρ0 δ1/δ2 ρs γp1/γp2 χ2(B/C, ¯p/p) Conventional 20 4.0 5.75 4.0 0.34/0.34 9.0 1.82/2.36 204.28,752.77 χ2− improved 20 4.0 5.75 4.0 0.34/0.34 9.0 1.82/2.33 265.09,438.753
Table 5.1: The parameters in two propagation models. D0 is in units of 1028 cm2· s−1, the break rigidities ρ0 and ρs are in units of GV.
synchrotron. The parameter τf,r is the time scale for fragmentation/radioactive decay of the cosmic-ray nuclei while they are interacting with interstellar hydrogen and helium.
Now we assume that the DM particle is Majorana fermions, which the particle is its own anti-particle. Hence, the primary source term from the annihilation of DM has the form:
where ρ is the DM density distribution function, hσvi is the thermally averaged cross-section, and dN/dE is the injection spectrum of anti-protons from DM annihilating into different SM final states(b¯b, u¯u and W+W− will be discussed in this work).
First we consider the so-called “conventional” diffusive re-acceleration model as the astrophysical background, then we are going to use another “χ2-improved” model, which fits the ¯p/p data better by using the frequentist χ2-test, in order to see the limitation without primary sources. The expression of χ2-test is defined as
χ2 =X
i
fith− fiexp2
σi2 , (5.3)
where fith, fiexp, and σi are the theoretical predictions, the central values of experimental data and errors of experimental data, respectively. The index i runs over all data point. A smaller χ2 value means the data is more fit.
The ¯p/p and the B/C ratio predicted by the conventional and the χ2-improved models with the latest AMS-02 data[12,13] are shown inFigure 5.1. Note that the predicted B/C
13
Figure 5.1: The predictions for ¯p/p and B/C in two propagation models with the latest AMS-02 data. The parameters are listed in Table 5.1.
ratio in both model are a bit higher for the energy below 10 GeV, but are consistent with the experiment data for higher energies. Since we don’t see any significant structure in the B/C ratio, the agreement between theoretical expectation and experiment is confirmed.
However, this is not the case when we look into the ¯p/p flux ratio data. If one wants to get a much better fitting curve, without introducing a new primary source, the B/C ratio gets worse, which is shown in Figure 5.2. Accordingly, a better strategy is to fix the B/C ratio, which can be done by fixing the propagation parameters, and we will pick the improved model as the background, then add the contribution from DM to see the improvement of the χ2 value.
Next we consider the contribution from three different DM annihilation channels b¯b, u¯u, and W+W−. Their injection spectra dN/dE are calculated using the PYTHIA 6.4.14[14]
package. Assuming the dark matter halo profile is isothermal expressed by
ρ (r) = ρ
rc2+ r 2
r2c+ r2, (5.4)
where the local DM density ρ = 0.3 GeV ·cm−3, the core radius rc= 3.5 kpc, and ρ = 8.5 kpc is the distance of the Sun from the galactic center.
14
Figure 5.2: The χ2-value contour shows a opposite behaviour for fitting the AMS-02 data.
The values of x-y axis are the break rigidity indexes γp1/γp2.
In Figure 5.3, it shows the best χ2-value cases for every final state. The order of the cross-sections are not the same comes from the very different injection spectrum, which represents how energetic the final state is. Interestingly, we scan the parameter space of the mass from 600 to 3000 GeV and cross-section from 1 ×10−26 to 3 ×10−24 cm3· s−1 in order to fit the peak around 200 GeV, and it turns out that the smallest χ2-value curve is not going to reach the highest data point. Although we do have some promising curves, which seem to fit the peak, they don’t fit other data points well with bigger χ2-values (two examples are shown in Figure 5.4). Since the χ2-value difference between the best χ2-value case and other promising curves is so little, we can only conclude that there are some possible masses and cross-sections around certain regions.
15
��->��
Figure 5.3: Twelve best cases for three different final states. The masses, cross-sections, and χ2-value are shown in Table 5.2.
Figure Mode Mass(GeV) Cross-section χ2-value
(a) b¯b 1200 1 ×10−24 390.692
Table 5.2: The attributes in each figure, which the cross-section is in units of cm3 · s−1. Every case has approximately 10% improvement in χ2-value compare to the background.
16
Figure 5.4: (LEFT) A DM to u¯u curve seems to reach the peak, but not the smallest χ2-value with 400 GeV and hσvi = 3 ×10−25 cm3· s−1. (RIGHT) Same story goes to the b¯b case with MDM = 1100 GeV and hσvi = 2 ×10−24 cm3· s−1.
17
Chapter 6 SUMMARY
We have seen the indications of DM from different scale of the Universe. Even though there are several theories claiming that the unexpected observations can be described without DM(i.e. MOND1), the story only goes if DM is included in every aspect. Besides lacking a DM candidate, the Standard Model has other issues (i.e. baryon asymmetry, the non-zero masses of neutrinos, etc.), thus the expansion of SM is greatly desired. To explore the essential of DM, we need to check its every footprint. In particular, we are trying to find some clues by studying the AMS ¯p/p data. Though we are not going to have a strong conclusion for the properties of DM, we do find that it is possible to improve the agreement to the AMS-02 ¯p/p data by introducing a DM contribution.
1Modified of Newtonian Dynamics[15]
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