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Dark Matter
Adrian Down
October 20, 2006
1 Review
1.1 Evidence for dark matter
In the previous lecture, we discussed some of the current evidence for the existence of dark matter. The most compelling piece of evidence currently comes from dynamics: luminous matter appears to be moving more quickly than can be explained from the amount of observed matter. Possible explanations are that there is some type of matter that has not yet been observed or that gravity deviates from general relativity. Currently, the best theories of modified gravity still require the existence of dark matter, although only hot dark matter, probably in the form of 2 eV neutrinos. However, all theories of modified gravity still have difficulties matching the observed spectrum of the CMB.
1.2 Dark matter problems
Last time, we saw that there are two problems with the amount of observed matter,
Ωm Ωb Ωluminous
Ωb Ωluminous requires the existence of some type of dark baryonic mat- ter. We discussed some possible candidates in the previous lecture, including MACHOs and various types of dead stars. The current best candidate is warm intergalactic gas. Ωm Ωb implies the existence of some type of exotic non-baryonic dark matter. The candidates most commonly mentioned are neutrinos.
2 Neutrino mass measurements
2.1 History
Neutrinos have been the natural candidate for hot dark matter since questions of the existence of dark matter arose. The most important parameter in determining the extent to which neutrinos can solve the nonbaryonic dark matter problem are the neutrino masses. Thus neutrino mass measurements are directly relevant to the non-baryonic dark matter problem.
Note. In the standard model of particle physics, neutrino masses are assumed to be 0, mostly for convenience.
2.2 Particle decays
The life times of particles that can decay into neutrino channels are sensitive to the mass of the various neutrino species. Current lab measurements of decay times place limits on the neutrino masses,
mντ < 18 .2 MeV mνμ < 190 keV mνe < 2 eV
These measurements are difficult, and it is unlikely that the limits on the masses of the μ and τ neutrinos will be on the order of an eV for a long time.
2.3 Neutrino oscillations
2.3.1 Quark analog
If neutrinos have mass, it should be possible for them to oscillate between types. This effect has been observed, although current measurements are sensitive only to neutrino mass differences, not to the neutrino masses them- selves. In the standard model of particle physics, the mass and weak eigenstates of a particle do not have to be identical. This effect has been observed extensively in the case of quarks. Quarks come in pairs, and the relation between the weak (primed) and mass (unprimed) eigenstates of the pairs can be written in matrix form,
d′ s′ b′
= V
d s b
Quantum mechanics The mixing of the generations of neutrinos is gov- erned by quantum mechanics. Suppose that at time t = 0, a pure νe state is produced. Writing this state in bra-ket notation,
|ν(0)〉 = |νe〉 = cos θ |ν 1 〉 + sin θ |ν 2 〉
At a time ∆t later, the state of this neutrino has evolved according to the propagator,
|ν(t)〉 = cos θ · e−ıE^1 t^ |ν 1 〉 + sin θe−ıE^2 t^ |ν 2 〉
where Ei =
p^1 + m^2 i. The probability of finding a νμ state at time t is given by the square of the projection of the final state onto νμ,
P (νe → νμ) = |〈νμ | ν(t)〉|^2
=
∣(− sin θ 〈ν 1 | + cos θ 〈ν 2 |)
cos θ · e−ıE^1 t^ |ν 1 〉 + sin θe−ıE^2 t^ |ν 2 〉
∣^2
Using the orthogonality of the basis states to cancel the cross terms,
P (νe → νμ) = cos^2 θ sin^2 θ
∣e−ıE^1 t^ − e−ıE^2 t
∣^2
sin^2 2 θ 2
[cos (E 1 − E 2 ) t]
= sin^2 2 θ · sin^2
(E 1 − E 2 )t 2
The energy difference can be simplified, assuming neutrinos are relativistic ,
E 1 − E 2 ∼
m^21 − m^22 E
=
∆m^2 c^4 E
Note. This is where the mass difference is introduced into the discussion of neutrino oscillations.
The probability of transition is thus,
P (νe → νμ) = sin^2 (2θ) · sin^2
∆m^2 · L E
where ∆m is in units of eV^2 , L is measured in meters, and E is measured MeV. In practical situations, energy can be measured fairly exactly. Thus any experimental measurements can simultaneously constrain both ∆m and sin^2 θ. The experimental details of this process will be covered in more detail in the student presentations later in the course.
2.3.3 Summary of 2006 measurements
From solar neutrino experiments plus the KamLand Japanese reactor exper- iment,
∆m^2 = 8. 0 +0 − 0 ..^64 · 10 −^5 θ =
33. 9 +2 − 2 ..^42
Note. ∆m is very small, and the large value of θ indicates that there is a large amount of mixing occurring amongst solar neutrinos.
From atmospheric measurements,
∆m^2 atm = (1.9 to 3.0) · 10 −^3 eV^2 sin^2 2 θatm > 0. 90
The large angle indicates that mixing is almost maximal. Based on the smallness of the mass difference, the current argument is that neutrino masses are likely very small. It is aesthetically undesirable that neutrino masses would be large but nearly degenerate, as required by the small mass difference.
Note. The large mixing angle observed for neutrinos contrasts with the case of quarks, in which case the mixing angle is small.
The existence of a non-zero neutrino mass difference indicates that neu- trinos do have mass, which is an exciting conclusion. However, these masses are probably very small. With such small masses, it is unlikely that neutrinos could be cosmologically significant. Neutrinos could be important if neutrino masses are large yet degenerate, however.
2.4 Supernova neutrinos
2.4.1 Theory
There are not yet strong mass limits on neutrinos from supernovae mea- surements. Stronger constraints would require a supernova explosion to take
The travel time is thus,
T =
d v
d c
m^2 ν c^4 E^2
The difference in arrival time between neutrinos and photons is related to the neutrino mass. Taylor expanding the expression for T ,
∆t = T − T 0 ≈
d c
m^2 ν c^4 2 E^2
Scaling the variables in convenient units,
∆t = 0.0257 sec ·
d 50 kpc
mν 1 eV
10MeV E
Thus the larger the neutrino mass mν , the larger the differences in arrival times between photons and neutrinos. This analysis has been carried out for the supernova 1987A. 12 events were observed at ∆t ∼ 12 and 8 events in ∆t ∼ 6. Based on these measurements, mν ≤ 10 − 20 EV.
2.5 Cosmology
Structure formation provides strong constraints on the neutrino mass. Re- call that neutrinos decouple at ≈ 1 second, when the temperature kT was about 1 MeV. The relic neutrino background should be related to the photon background via,
Tν, 0 =
)^13
Tγ, 0 = 1.945 K
which implies a number density per species of,
nν, 0 = 113 cm−^3
We showed on a previous homework that the total mass density of neutrinos can be calculated as a sum over the masses of individual neutrino species,
Ων h^2 =
i mi 93 .5 eV
We have seen previously that a matter-only universe is inconsistent with the age of some observed objects in the universe. Thus Ων must be much less than 1, which provides a powerful constraint on the mass of neutrino species to be less than about 40 eV. The current WMAP constraints based on structure formation imply that
∑
i
mi ≤ 2 eV
With this mass, the neutrino density Ων is insufficient to account for the gab between Ωm and Ωb. Thus most of the nonbaryonic dark matter discrepancy must come from cold dark matter.
3 Cold dark matter
3.1 WIMPs
Currently, very little is known about cold dark matter. This type of dark matter is called “cold” because it is heavier, and thus does not have a large amount of momentum. The generic name used to refer to cold dark matter is WIMPs, which stands for weakly interacting massive particles. “Weakly interacting” refers to the fact that these particles cannot couple strongly to photons, otherwise WIMPs would have already been observed. “Massive” refers to the fact that WIMPs are cold.
Note. WIMPs are non-baryonic dark matter, whereas MACHOs are baryonic dark matter. Don’t confuse the two.
3.2 Neutralinos
The current best WIMP candidate particle is the neutralino, which is the lightest hypothesized super symmetric particle. We will not pursue the par- ticle physics of super symmetry in detail. However, the general idea of super symmetry is that each particle in the standard model has a partner that has not yet been observed. These super symmetric particles must be very massive to avoid being detected yet in particle accelerators.
4.2 Linear perturbation theory of gravitational insta-
bility
4.2.1 Motivation
As with most perturbation analyses, we begin with the linear regime for sim- plicity. This theory is extremely important and is the basis for all linear perturbations used in cosmology. The basic idea comes from the Jeans in- stability, first proposed by James Jean an 1902. The theory employs a static medium filled with non-relativistic fluid with mass density ρ, pressure P , velocity v, and gravitational potential φ.
4.2.2 Continuity equation
The basic equations governing this fluid come from fluid dynamics. The first equation is the continuity equation, which expresses mass conservation,
∂ρ ∂t
4.2.3 Hydrodynamic force
The second equation is a force law, analogous to Newton’s law in the case of a fluid,
∂v ∂t
ρ
∇P − ∇φ
Note. • This equation is known as Euler’s equation in the non-viscous case. It is also called the Navier-Stokes equation.
- The left side of the above equation expresses the time derivative of the velocity, as the expression for the time derivative is more complicated in the case of a fluid,
d dt
∂t
i
dxi dt
∂xi
∂t
- The first term on the right side of the above equation is a pressure force.
- The second term on the right represents the force of gravity. Many terms can be added on the right side of the above equation repre- senting different physical effects. For example, viscous and magnetic forces would be represented, respectively,
Fv = ν∇^2 v FB = −
ρ
8 π
∇B^2
In our universe, the fluid can be treated as inviscid, and the magnetic field is negligible to a good approximation.
4.2.4 Poisson’s equation
The third important equation is Poisson’s equation, which relates the mass density to the gravitational potential,
∇^2 φ = 4πGρ
4.2.5 Equation of state
Note. • We have four variables, but only three equations. To deal with this problem, we will introduce the equation of state to relate pressure to density.
- We will see that all of these equations are velocity moments of the more general Boltzmann equation.
- These equations assume that all particles are interacting as in a fluid. However, we saw previously that this is not a good approximation at all times in the history of the universe.
