Lecture 12
Dark Matter
Part II
What it could be and what it
could do
Theories of Dark Matter
What makes a good dark
matter candidate?
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Charge/color neutral (doesn't
have to be though)
Heavy
We know KE ~ keV
CDM ~ GeV or heavier
Non-baryonic
Fall out of larger theory
Natural production
mechanism
e.g. WIMP miracle
More theories than you can count!
Let's discuss a few of them
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Old News: MACHOs
Massive Compact Halo Objects
● Jupiter-like planets, brown dwarf stars
● Extra “normal” matter that doesn't emit light
But MACHOs are baryons!
● Doesn't this disagree with WMAP?
● MACHO searches were conducted about
the same time that the CMB
measurements were made
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Look for MACHOs via gravitational lensing
Micro-lensing of starlight from within the
milky way
Look for brightening of star as MACHO
passes in front of it
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Galactic survey to look for MACHOs
in the milky way
Some MACHOs were found
Not enough to constitute DM
Consistent with WMAP result that
DM is non-baryonic!
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Sterile Neutrinos
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We know about 3 types of neutrinos
Could there be a fourth type?
The full answer will be discussed in neutrino lectures
Short answer, yes, but it must be sterile!
● Doesn't participate in weak interactions
● Only accessible via oscillations
Usually heavy (~ keV)
Not quite cold, but warm DM candidate
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Axions
Remember CP violation?
● C = charge congugation
● P = parity (invert coordinates, like looking in mirror)
● CP symmetry means that under inversion of charge and
parity, physics is unchanged
● Does an electron behave the same as a positron when
viewed in a mirror?
● CP violation: processes which do not obey this symmetry
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CP violation observed in weak interactions
Never observed in strong interactions
But remember Noether's theorem:
Symmetry
Conserved quantity
No such symmetry in strong interaction
Strong CP problem
Add axial symmetry to strong interaction
Resulting particle is the axion
Would solve strong CP problem
Could also be DM
Huge parameter space for axion coupling and mass
Subset gives correct DM density
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ADMX Experiment
● Search for axion dark matter
● No discovery yet
● Not ruled out either
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WIMPs
Recall the WIMP Miracle
● Weak-scale cross section gives correct
abundance
● Many, many theories give WIMPs
● Extensions to the SM often have new
physics just beyond our reach
Electroweak scale
● New particles and new interactions
● New forces or extensions of existing ones
● Natural framework for CDM
● Heavy (e.g. cold) particles
Scaling of WIMP miracle:
● Relic density roughly scales with
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Smaller cross section but larger mass gives
same relic density
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SUSY
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Remember our old friend
SUSY
Doubling of SM
Fermion – Boson
symmetry
Lightest SUSY Particle
(LSP)
Naturally neutral
Neutralino:
SUSY is by far the most popular framework for physics beyond the SM, and DM
A lot of flexibility
Answers many shortcomings of SM
By no means the only possibility for WIMP DM
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Composite Dark Matter
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DM could be composite
Made up of constituent particles
Like baryons consist of quarks
One example:
● DM consists of 4 fundamental particles
● Each carries both weak and electrical charge
● But the DM particle is electrically neutral
● Couples through Z, H exchange
Many other models:
● Different numbers of constituents
● Different properties
● EM, weak, strong interactions
● New interactions
● Always globally neutral (like neutron)
● Come from adding a new symmetry to SM
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Mirror Dark Matter
Mirror copy of normal particle physics
All known particles have a copy in the mirror sector
Same physics in the mirror sector as normal sector
The two sectors are connected by a photon mixing
Consistent with WIMP miracle
High density allows freezeout as with other WIMPs
Mixing is small (ε ~ 10-9)
Characteristics:
More dynamic physics available in DM halos
Bullet cluster and non-bullet cluster like collisions
Dominantly an EM interaction
But strongly suppressed by ε
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Compact Dimensions:
Kaluza Klein Particles
Compact extra dimension
Periodic boundary
Think of a loop
Compact, ~ TeV-1 (10-21 m)
Quantized excitations
Shows up as effective mass
~ TeV particles
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In Kaluza Klein theory, the
lowest mode is stable
Invoke a symmetry to protect
KK number
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Orthogonality:
New dimension (w) orthogonal to x, y, z
Welcome to the fourth dimension
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WIMP Miracle and DM Detection
The WIMP Miracle predicts
three types of interactions
DM annihilation
DM production
DM scattering
We look for all three
experimentally!
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WIMP Scattering
Let's start with a few assumptions:
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Cold Dark Matter exists and consists
of SUSY WIMPs with m ~ 100 GeV
The WIMP miracle tells us DM should
interact with normal matter!
Our galaxy is immersed in a WIMP
halo with density ~ 0.3 GeV/cm3
The WIMP velocity distribution is
roughly understood (details later)
WIMPs and matter have weak scale
interaction
How do we determine the interaction rate in a WIMP detector?
We can follow general scattering theory
New physics only shows up in the cross section (in most cases)
→ We can ignore the details, and calculate the rate for a given
cross section
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Scattering Theory
In quantum mechanics, you will study scattering
theory
Probability (or rate) of interaction
In general, it can be written in this form
Two things to deal with:
● Velocity integral
● Differential cross section
R = scattering rate
ρ0 = 0.3 GeV/cm3 is the local dark matter density
mx and mN are the WIMP and Nucleon masses respectively
v is the WIMP speed
f(v) is the 1D WIMP speed distribution
S is the elastic WIMP-nucleon cross section
q is the momentum transfer
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Some Useful Substitutions
Reduced mass:
Energy transferred:
At low momentum transfer:
σ0 is the cross section at zero momentum transfer
F2(Q) is the nuclear form factor
Integrating over all possible velocities,
the differential rate can be written as:
Three terms to deal with:
● Velocity integral
● Cross section
● Form factor
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Details of the Velocity
Minimum velocity: bound by kinematics
Two cases for maximum velocity:
Simple model: vmax = ∞
Realistic model: vmax = vesc
WIMPs greater than vesc are not gravitationally bound
Different measurements exist for the
escape velocity in the Milky Way
496 < vesc < 655 km/s
Earth-Sun motion:
The sun moves around the galactic center at
v0 = 220 km/s
The earth moves with or against the sun motion
Annual modulation of relative WIMP velocity
distribution
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Simplest Model:
Ignore vesc and vearth
Maxwellian distribution:
Velocity integral:
What does this mean for the WIMP scattering rate?
Consider F2(Q) = 1 (true for light WIMPs):
General feature:
Exponential Energy Spectrum!
This is a global feature, all additional terms only modify this slightly
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More Realistic Model:
Include Earth-Sun Motion
Recall:
The 1D Boltzmann distribution has two terms
(derivable from 3D distribution)
With this form and still ignoring the escape
velocity, the velocity integral is:
General feature:
Exponential Energy Spectrum!
This is a global feature, all additional terms only modify this slightly
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The Full Solution
Re-evaluating the previous integral with finite vesc
And the differential rate can be written as:
vmin is a function of Q
vE is a function of cos(t)
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Features of the Velocity Integral
Exponential behavior predicts high rate at low recoil energy
→ Experiments must be sensitive to few keV
Earth-Sun motion creates modulation in rate
Modulation in total number of events
May ultimately allow for probing our own DM Halo
Escape velocity truncates rate at high E
More complicated velocity distributions can be considered
(streamers, motion or structure in Halo, etc)
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Form Factor
Describes how a wave overlaps with nuclear structure
How much nuclear structure do you see
Recall the DeBroglie wavelength
Long wavelength
sees nucleus as
one “blob”
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Short wavelength
resolves nuclear
sub-structure
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Computing the Form Factor
The nuclear form factor is the Fourier transform
of the matter density in the nucleus
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Need to know matter density inside
nucleus
Nuclear physics
Many different form factors used
Most agree to a large extent
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Example of Form Factor
Woods-Saxon form factor
Treat the nucleus as a hard, uniform sphere, but with blurry edges
j1(x) is a spherical bessel function
q is the momentum transfer
R1 is related to the nuclear radius R
s is the “skin depth” of the nuclear
edge
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Features of the Form Factor
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Suppresses high
momentum exchange
Poles at certain
momentum exchange:
● Destructive interference
● Form factor → 0
● Rate → 0
Goes to 1 as q goes to 0
No effect when no
resolution of nuclear
structure!
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Cross Section
The cross section at zero
momentum transfer is:
# protons
# neutrons
Proton coupling
Neutron coupling
Let's make an assumption: fp = fn
Maybe that's not too crazy,
we're looking for SUSY WIMPs
Ignoring form factor effects, the rate should scale with A2
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The Final Event Rate
Plugging in all the numbers, one gets a
differential rate for each WIMP mass,
cross section and target material
Features of this plot:
Differential Rate Unit (dru)
Events / kg / keV / day
A2 dependence → Heavy
target is better (at low E)
Form factor → large
nucleus has pole at low E
Incredibly low rate!
(few events / year)
→ experimental challenge
Mχ = 100 GeV, σ = 10-44 cm2
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