DARK MATTER

Fig1: A GALAXY

In astronomy and cosmology, dark matter is a type of matter hypothesized to account for a large part of the total mass in the universe. Dark matter cannot be seen directly with telescopes; evidently it neither emits nor absorbs light or other electromagnetic radiation at any significant level.

Evidence

1. Galactic Rotation:

Much of the evidence for dark matter comes from the study of the motions of galaxies. Consider our Milky Way galaxy, which is a spiral galaxy with Sun in one of the spiral arms about 2/3rd of the way from the centre. The bright central region contains most of the mass (visible). If we apply Kepler’s third law to the rotation of star we get,

V= (GM/r) ½

Where, G is the gravitational constant

M is the mass at the centre of galaxy

R is the distance of the star from centre of galaxy

V is the tangential velocity of the star.

So, we expect V to decrease as the r increases. But the experimental fact suggests that V almost remains constant with increasing mass.

Fig 2: Velocity of star w.r.t the distance from the galactic centre

“A” is the predicted and “B is the observed.

2. Gravitatinal Lensing:

When a light passes through a distribution of masses, it bends as predicted by Einstein’s theory of General Relativity. When we observe light from a background galaxy, the mass distribution of the galaxy in between can be found out by the bending of light which passes through it. Experimental evidences show that the visible matter does not account for the observed bending.

Fig3: Bullet Cluster

In this figure, the blue region corresponds on the dark matter as inferred from gravitational lensing. The red region corresponds to the matter emitting Xrays and the white to the matter emitting optical rays.

As it can be seen, during the collision of two galaxies, the spherical shape of the dark matter is not disturbed. Thus it could be Weakly Interacting Massive Particles (WIMP).

Explanation of the discrepancies.

1. Newton law’s of gravitation does not apply at such large scale.2. Additional Matter exist which are not visible but which exerts gravitational force

necessary to account for the data. (Dark Matter)

Laws of gravitation as stated by Newton has helped us to understand various phenomena which give the exact predictions, so there is little chance that this could all be wrong.

So the strong possibility seems to be Dark matter.

Candidates for Dark Matter:

1. Massive Compact Halo Objects. (MACHOs)2. Neutrinos.3. WIMP, such as predicted by supersymmetric theory (as discussed below)

Astronomical surveys did not find enough evidence of MACHOs which led to eliminating that option.

After the discovery of neutrinos oscillation which proved that they have tiny masses, they were considered as a likely candidate. But Neutrinos travel at relativistic speed, thus it would have high thermal energy which forms a class of dark matter called “HOT” dark matter, these cannot explain the binding of cluster of galaxies.

WIMPs move with non-relativistic speed at the time of freeze-out. Heavier particles decay into the light, stable SUSY particle called Neutralino, a neutral fermion which arises from linear combination of photino, Zino and two higgsinos(described in pages below). Since these particles don’t form atoms, they interact only via weak forces. They seem to be the most likely candidate.

Brief review of Particle Physics

The idea that all matter is composed of fundamental constituents dates back to 6th century BC. The quest of physicists since then was to find these constituents. During the early 19th century, atoms were discovered and they were thought to be the building blocks of matter and that it was indestructible. This was disproved by the works of various scientists who discovered the protons, neutrons and electrons. But today in the beginning of 21st century, (with all the particle accelerators and colliders) we have found hundreds of particles which are all unstable. It was thus necessary to see whether this large zoo of particles can be explained in terms of smaller particles. In this way a Standard Model of particle physics gradually developed.

Forces of nature:

1. Strong force – Nuclear binding2. Weak force – Radioactive decay3. Electromagnetism – Interaction of charged particles.4. Gravitation – Free fall.

Fig 4: Fundamental Forces

CLASSIFICATION OF ELEMENTARY PARTICLES.

Based on spin number:

1. Fermions: They are particles with half integer spin which are subject to Pauli’s exclusion principle.

Eg: Electron, proton, neutron.

2. Bosons: They are particles with integral spin which obey Bose-Einstein statistics. Eg. Photon,gluon,w &z boson,graviton

Fermions are further divided into two elementary particles

1. Quarks: These are fundamental constituents of Hadrons (pg 5) .These are of six types: up,down,strange,charm,bottom,top.

Fig 5: Quarks

2. Leptons: They do not interact via strong interactions. They form 3 generation which has similar properties but only increasing mass.

First generation: Electron and Electron neutrino. Second generation: Muon and muon neutrino. Third generation: Tau and tau neutrino.

The heavier muons and tauns will rapidly undergo particle decay. All neutrinos have negligible mass and zero* charge.

Fig 6: Leptons

*as previously thought.

Hadrons: These are composite particles of quarks.

a) Baryons: They have 3 quarks.

Eg: Protons u(2/3), u(2/3), d(-1/3)

Neutron d(-1/3), d(-1/3), u(2/3)

Charges are mentioned in the bracket.

Mesons: They have a quark and an anti quark pair.

Eg: Pion(u,d antiquark) and Kion (u,s antiquark)

Conservation laws

In addition to the charge, momentum and energy conservation laws, in particle physics we also have :

1. Baryon number conservation (conserved by all interactions: strong, weak and EM )All the baryons have baryon no. +1 and their antiparticles have baryon no. -1All the mesons have baryon no. 0Eg:- + p+ no + - + + 0 1 1 0 0

2. Strangeness conservation (conserved by strong and EM interactions but not by weak interactions)Strange baryonsS= -1(∑+,∑0,∑-,λ)S= -2(cascades)Strange mesonsS=-1(k+,k0)S=+1(k-,k0 antiparticle)Therefore pair of strange particles can be produced by collision of non strange particles.Eg. - + p+ λ + k0

B 0 1 1 0S 0 0 -1 +1Strange particles decay via weak interactions which does not conserve strangenessEg. λ - + p+ B 1 0 1 S -1 0 0

From, study of all these particles, the Standard model was developed.

THE STANDARD MODEL

Fig 7 : Table of the standard model

Standard model mainly consists of two types of fields.

1. Matter field: These are the particles which form matter. Quarks (Violet) and Leptons (Green) belong to this category.

2. Gauge field: These are also called the force carriers. Matter interacts through the exchange of these particles and experiences the different kinds of forces.

Strong interactions: Gluon (g) Weak Interactions: W+, W-, Z Electromagnetism: Photon (γ) Gravitation: (graviton*)

Fig 7:Types of Interactions

Except for photon and gluon, all other particles have mass. This is attributed to their interaction with the Higgs field. When we write down the interactions of various particles in the table, it turns out that the Higgs Boson do not couple with photon and gluon.

The search for this Higgs boson has been one of the primary goals of Large Hadron Collider (LHC) which is a particle accelerator. According to the latest outcome, they might have found it.

(*Yet to be discovered.)

Lepton Number Conservation.

Initially: It was observed that when an electron neutrino decays, electron is one of the products. And when a muon neutrino decays, muon is one of the products. So it was assumed that each flavor has a separate lepton number and they should also be conserved in a reaction.

Eg:e + n p + e-

le 1 0 0 1 lµ 0 0 0 0

n p+µ-

le 0 0 0 0 lµ 1 0 0 1

A strange quantum mechanical phenomena

Neutrino OscillationsThe standard solar model gives the flux density of the electron neutrino that would reach the earth. But it was found that, only one third to one half of predicted numbers of electron neutrinos were detected. Then it was discovered that neutrinos oscillate from one flavor to another (Eg: electron to muon)

Fig 8: Neutrinos from the Sun

By reverse beta decay process, the proton and an electron combine to give a neutron with electron neutrino as one of the products. This electron neutrino interacts with the matter (neutron) and electron and proton are produced. Detectors on earth detect the electron from this interaction. Their mathematical explanation is as follows.

There are two different types of Eigen States.

Flavor Eigen states: e

Mass Eigen states:

In weak decays, the flavor Eigen states are produced. It does not have a definite mass. They are represented by Dirac bracket notation.

|e> =cos∅ |> +sin∅ |>|> = -sin∅ |> + cos∅ |>

Approximation that there are only two flavors of Neutrino. There are actually three.

The probability of a electron neutrino at t=0 converting into a muon neutrino at time t is given byP(e, 0; , t) = |<e(0)| (t)>|2 = 4 (cos∅ sin∅ sin∝)^2

Where = ((2πm22 – m1

2)C3t)/ 4ph

Note: 1) If m1=m2=0 then the probability will be zero which contradicts experiments. 2) We can only determine m1- m2 and never the definite mass.

Neutrinos oscillations have been observed and from that the difference in mass calculated. These were thought to be a likely dark matter candidate but as explained in the beginning, they are now overruled.

Even though the standard model explains most observed phenomena, it does not have any Dark matter candidate. Supersymmetric model was established. And as discussed below, this has dark matter candidates due to symmetry.

Supersymmetry (SUSY)

Corresponding to every Fermion in the standard model, there must be a Boson and vice versa. They are called super partners. Supersymmetry is broken in nature, i.e all super partners are heavier than SM particles.

Lepton (Spin ½) Slepton (spin 0)Quark (Spin ½) Squark(Spin 0)

Similarly,

Gauge Boson (spin 1) Gaugino (spin ½)Higgs boson (Spin 0) Higgisnos (spin ½).

Fig 8: Standard Model and Supersymmetric particles

χ01,2,3,4 are the Neutralinos which are a mixture of Zino, Photino and neutral higgsinos

χ01 is the lightest stable particle. This stability is due to certain symmetry (R parity). These interact only

via gravitation and weak forces. Thus, it is a possible candidate for dark matter.

Dark Matter from Big bang:

Just a few seconds after the big bang, the temperature was very high, hence kT was large. Large unstable particles decay into lighter particles and also collision of lighter particle also produces heavy particles. Equilibrium is achieved. Under this equilibrium, the number of heavy particles is proportional to the Boltzman constant e-m/kT

As the universe expands, it cools down and hence kT decreases. Thus, the heavy particles are not produced, and the existing unstable ones decay. But consider weakly interacting stable particles. As the universe expands, the annihilation may become rare as its density is low. Thus there is a departure from the equilibrium and they remain forever. Moreover, if they do not interact electromagnetically, they become a very strong dark matter candidate. Neutralino is one such particle.

Fig 9: Big bang expansion and dark matter

It shows the expansion of the universe and the point at which the W-map collaboration calculated the relic density of dark matter

Dark matter producing interactions:

χ01 + χ0

1 SM particles (Annihilation)χ0

1 + A SM particles (Co- Annihilation)

Where A is another SUSY particle whose mass is comparable to that of Neutralino.

The dark matter relic density is calculated by numerical solution of the Boltzsman equation. The W-map collaboration has noted the observed values as 0.09-0.13 h2. Where, h is the Hubbles constant.

By varying parameters like the mass of Slepton and Neutralino, the relic densities are calculated by a standard package. This is then compared with the calculated value. Neutralino-Neutralino Annihilation :

Two Neutralinos can annihilate into SM particles and this process contributed to Dark matter relic densities. This occurs via the exchange of a virtual Slepton. It turns out that this relic density (Ω) increases as the annihilation cross section (σ) decreases.

As we increase the Mass of Slepton, σ decreases and hence relic density Ω increases. This is because in this interaction, Slepton is a virtual particle which disobeys the law of conservation of energy, with increase in mass, this process becomes more improbable.

1201301401501601701801902000

0.020.040.060.080.1

0.120.140.16

Omega at M1=47Omega at M1=57Omega at M1=45

Mass of slepton M_eR

Ω

Graph 1: variation of Ω with M_eR keeping M1 constant

As seen from the figure, for each mass of Neutralino, there is a range of Slepton mass in which Ω falls within the experimentally observed range.

Z- Resonance

When the mass of Neutralino is approximately half of Z boson (91Gev) Resonance occurs. This leads to Neutralino annihilate into the real Z boson which decays into fermion-antifermion pair. At resonance, Ω decreases. The quark-antiquark pair dominate over e-E and m-M.

Higgs Resonance.

When the mass of Neutralino is approximately half of Higgs. Resonance occurs. The Neutralino annihilate to into Higgs which decays into fermion-antifermion pair. During resonance, Ω decreases.

Top quark cannot be produced as it is heavier than higgs (172 Gev) . And so the next heaviest particle which is the bottom quark dominates.

Also, the fraction of bB is much larger in Higgs resonance than in Z resonance. (Graph 3)

As we increase the mass of Neutralino, σ increases and Ω decreases. This is because, with increase in mass, we have higher energy (E=mC2) and hence more annihilation.

43 45 47 49 51 53 55 570

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

M_eR = 120

M_eR=150

M_eR = 200

M1

Ω

Graph 2: variation of Ω with M1 keeping M_eR constant

As it can be seen, when M1 is approximately half of that of the mass of Z boson, resonance effect takes place. σ increases and thus Ω decreases. Also, when M1 is half of that of Higgs boson, Higgs resonance effect takes place.

41 43 45 47 49 51 53 55 57 59 610

20406080

100

Z and Higgs resonance

M1

% o

f bB

Graph 3: Change in contribution % of bB wrt M1

In the above graph, 2 peaks are found. One at M1=47 and the other at M1= 59. It is due to Z and Higgs resonance respectively.

Co annihilation:

When the Neutralino interacts with other SUSY particles of comparable masses ( like,Sneutrino) the pair annihilates and standard model particles are produced. The other particles could be Chargino, Stop or Sneutrino.

When the Sneutrino mass becomes smaller than Neutralino mass, then Ω drops. As then, neutralino is no more the LSP. With increase in Slepton mass, sneutrino mass increases.

CONCLUSION

Supersymmetry (or SUSY) is an attractive concept because it offers a solution to the "hierarchy problem" of particle physics, provides a way of unifying the strong and electroweak forces, and even contains a dark-matter particle. An important prediction of the theory is that every known particle has at least one superpartner particle – or "sparticle". The familiar neutrino, for example, is partnered with the yet-to-be discovered sneutrino. These sparticles are expected to have masses of about one teraelectronvolt (TeV), which means that they should be created in the LHC.

A possible Dark Matter candidate is the Neutralino. As seen above, we have obtained the dark matter relic density by varying parameters. For a range of these values, the relic density falls within the observed range. The search for these particles in supersymmetry is carried out at the LHC.

It would be interesting to see whether the discovered particles have mass which can explain the dark matter or else, we will have to look for alternative candidates.