Cosmology 566: Class 6a Dark Matter

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Cosmology 566 Class 6a Dark Matter
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1. Born Dark, and Proud of it Neutrinos
When are species in Equilibrium?
The Most important equation in cosmology (BBN,
DM,CMB) Boltzmann equation for expanding
Universe
Evolution of species
In T.E. nneq -gt n1/R3
But, once
Decoupling! Dist. Fn. Not TE value
massless
i.e. Not
massive
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Instead
Massless
Before D
After D
!!!
Prove
Massive
Before D
After D
Our existence! Neutrons..
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1. Born Dark, and Proud of it Neutrinos
FINALLY Neutrinos Recall SM SU(3) x SU(2)
x U(1) 3 generations left handed particles
Note no right handed neutrino?
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1. Born Dark, and Proud of it Neutrinos
FINALLY Neutrinos Recall SM SU(3) x SU(2)
x U(1) 3 generations left handed particles
Note no right handed neutrino? (Hence 2
helicity states instead of 4?
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1. Born Dark, and Proud of it Neutrinos
Note no right handed neutrino? (Hence 2
helicity states instead of 4?
Recall for massless particles helicity
chirality
Hence, left handed neutrinos and right handed
antineutrinos are the only particles in the
radiation bath in the SM. Note that two different
mass terms possible
i.e. 4 components or 2
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1. Born Dark, and Proud of it Neutrinos
Weak interactions ?W GF 2E2T2 for Tgtgtm Hence
n lt ?W vgt T5
Hence, for T -gt inf. n lt ?W vgt gtgt H and n?
n?neq
Specifically
Hence W.I. decouple at T MeV gtgt m?
Hence, following that, n? n?
How to determine neutrino remnant number density?
Entropy conservation
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1. Born Dark, and Proud of it Neutrinos
Recall that for particles in thermal eq S gs
R3T3 remains constant. Also since TDgtgtmn , after
decoupling Tn1/R
Now, photons and electrons remain in thermal
equilibrium, as electromagnetic reactions fast
enough.. At 3T .5 MeV, ee- begin to annihilate
(in equilibrium), producing photons, so that n(e)
exp(-M/T) after that. Thus, the entropy
originally stored in electrons is dumped into the
photon gas (but not the neutrino gas). We
have (gs R3T3)before (gs R3Tg3)after
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1. Born Dark, and Proud of it Neutrinos
(gs R3T3)before (gs R3Tg3)after Before gs
2 (g) 4 (ee-) x 7/811/2 After gs 2 (g)
Hence (Tn3/Tg3) 4/11 Thus (nn nn) 3/4 x
4/11 ng115 cm-3
THE FIRST DARK MATTER!
Hence if mn30 eV Wn 1 (h 75)
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2. Topological Defects
i.e. Magnetic Monopoles Dirac1931
g1/(2e) for q.m. to be consistent.
Explain Quantization of Charge? Non Abelian
(t Hooft Polyakov) When any semi-simple group
(i.e. SU(5) ) breaks to a subgroup containing an
explicit U(1) factor -gt magnetic monopoles with
charge g1/e. Dirac explain quantization of
charge! Monopoles B-gt0 inside (additional scalar
field f-gtr -map, say SO(3) -gt SO(3) ) and
singularity removed..
(Singular r0, And gauge string At qp)
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2. Topological Defects
Kibble Mechanism
Existence of an HORIZON! ct
Symmetry breaking phase transition, say order
parameter a scalar field ??that gets a VEV..
If vacuum manifold non-trivial field can get
different VEVs in different causally disconnected
regions If non-trivial closed curve in group
space spanned in physical space, a defect is
formed
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SU(5) Tc1016 GeV
1 Defect/horizon volume?
Prove
Calculate subsequent Number density by
Annihilation (see next Section.. Negiligible)
Overclose the Universe if mdefect is too big, or
horizon volume too smalli.e. T too large
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Dark Matter Achievers WIMPS
Prototype VERY HEAVY NEUTRINOS (Mgtgt 2 MeV)
Lee-Weinberg Calculation
Q If neutrinos are still interacting when the
temperature Falls below their mass, what happens?
Rewrite Boltzmann Equation
Define
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Dark Matter Achievers WIMPS
Now, during radiation dominated regime
Hence
Now to calculate neutrino annihilation cross
sections
A straightforward and illuminating but tedious
calculation Gives
For ann. into NA species
For Tltltm pltltm, and Em, thus
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Dark Matter Achievers WIMPS
Hence
Analytic approx Freeze-out occurs when
annihilation Rate dilution rate! (think about
this)
(1)
f0 equil. value at freezeout
ie
in equilibrium
Now
Thus, solving (1) for x ltlt1
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Dark Matter Achievers WIMPS
Thus
From then on
w bdry cond f(xf)f0(xf)
Hence
Thus, finally
(Prove)
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Dark Matter Achievers WIMPS
Note from above, for weakcross section, and m 1
GeV xf 1/20
Hence
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Massive Weakly Interacting Particle a Natural
Candidate for Dark Matter!
Unfortunately, Massive Neutrinos (lt1/2 Mz are
Ruled out by LEP!
Note what about a Majorana particle?