Particle PhysicsParticle Physicsand Cosmologyand Cosmology

Dark MatterDark Matter

What is our universe What is our universe made of ?made of ?

quintessence !fire , air,

water, soil !

Dark Energy Dark Energy dominates the Universedominates the Universe

Energy - density in the Energy - density in the UniverseUniverse

==

Matter + Dark EnergyMatter + Dark Energy

25 % + 75 %25 % + 75 %

Composition of the Composition of the universeuniverse

ΩΩb b = 0.045= 0.045

ΩΩdmdm= 0.225= 0.225

ΩΩh h = 0.73= 0.73

critical densitycritical density ρρcc =3 H² M² =3 H² M² critical energy density of the critical energy density of the

universe universe ( M : reduced Planck-mass , H : Hubble ( M : reduced Planck-mass , H : Hubble

parameter )parameter )

ΩΩbb==ρρbb//ρρcc

fraction in baryons fraction in baryons energy density in baryons over energy density in baryons over

critical energy densitycritical energy density

~60,000 of >300,000Galaxies

Baryons/AtomsBaryons/Atoms

DustDust ΩΩbb=0.045=0.045 Only 5 Only 5

percent of percent of our our Universe Universe consist of consist of known known

matter !matter !

SDSS

Abell 2255 Cluster~300 Mpc

Ωb=0.045

from nucleosynthesis,cosmic background radiation

Abell 2255 Cluster~300 Mpc

Matter : Everything that clumpsMatter : Everything that clumps

Dark MatterDark Matter ΩΩmm = 0.25 total “matter” = 0.25 total “matter” Most matter is dark !Most matter is dark ! So far tested only through gravitySo far tested only through gravity Every local mass concentration Every local mass concentration

gravitational potentialgravitational potential Orbits and velocities of stars and Orbits and velocities of stars and

galaxies measurement of gravitational galaxies measurement of gravitational potential potential

and therefore of local matter distributionand therefore of local matter distribution

gravitational lens , HST

ΩΩmm= 0.25= 0.25

Gravitationslinse,HST

Gravitationslinse,HST

Abell 2255 Cluster~300 Mpc

Matter : Everything that clumpsMatter : Everything that clumps

ΩΩmm= 0.25= 0.25

spatially flat universespatially flat universe

theory (inflationary universe )theory (inflationary universe )

ΩΩtottot =1.0000……….x =1.0000……….x

observation ( WMAP )observation ( WMAP )

ΩΩtottot =1.02 (0.02) =1.02 (0.02)

ΩΩtottot = = 11

ΩΩtottot=1=1

Dark EnergyDark Energy

ΩΩmm + X = 1 + X = 1

ΩΩmm : 25% : 25%

ΩΩhh : 75% : 75% Dark Dark

EnergyEnergyh : homogenous , often ΩΛ instead of Ωh

dark matter candidatesdark matter candidates

WIMPSWIMPS

weakly interacting massive particlesweakly interacting massive particles

AxionsAxions

many others …many others …

WIMPSWIMPS

stable particlesstable particles typical mass : 100 GeVtypical mass : 100 GeV typical cross section : weak typical cross section : weak

interactionsinteractions charge : neutralcharge : neutral no electromagnetic interactions , no no electromagnetic interactions , no

strong interactionsstrong interactions seen by gravitational potentialseen by gravitational potential

bullet clusterbullet cluster

How many How many dark matter particles dark matter particles

are around ?are around ?

cosmic abundance of relic cosmic abundance of relic particlesparticles

early cosmology : present in high early cosmology : present in high temperature equilibriumtemperature equilibrium

annihilation as temperature drops annihilation as temperature drops below massbelow mass

annihilation not completed since annihilation not completed since cosmic dilution prevents interactionscosmic dilution prevents interactions

decoupling , similar to neutrinosdecoupling , similar to neutrinos relic particles remain and contribute relic particles remain and contribute

to cosmic energy densityto cosmic energy density

cosmic abundance of relic cosmic abundance of relic particlesparticles

rough estimate for present number density n :rough estimate for present number density n : n an a3 3 constant after decouplingconstant after decoupling n/s approximately constant after decouplingn/s approximately constant after decoupling compute nacompute na3 3 at time of decoupling at time of decoupling roughly given by Boltzmann factor for roughly given by Boltzmann factor for

temperature at time of decoupling temperature at time of decoupling

and zand zdecdec

cosmic abundance of relic cosmic abundance of relic particlesparticles

number density for WIMPS much number density for WIMPS much less than for neutrinos less than for neutrinos

(Boltzmann factor at time of (Boltzmann factor at time of decoupling )decoupling )

ρρ = m n = m n large mass : large mass : ΩΩm m > > ΩΩνν possible possible

computation of time computation of time evolution of particle evolution of particle

numbernumber

central aspects :central aspects : decoupling of one species from decoupling of one species from

thermal baththermal bath other particles in equilibriumother particles in equilibrium Boltzmann equationBoltzmann equation occupation numbersoccupation numbers

book : Kolb and Turnerbook : Kolb and Turner

occupation numbersoccupation numbers

ρρ

ininequilibriumequilibrium

Boltzmann equationBoltzmann equation

squared scattering amplitudesquared scattering amplitude

phase space integralsphase space integrals

dimensionless inverse dimensionless inverse temperaturetemperature

dimensionlessdimensionlesstime variable time variable in units ofin units ofparticle mass m particle mass m

m : particle massm : particle mass

small x : particle is relativistic x<3small x : particle is relativistic x<3large x : particle is non-relativistic x>3large x : particle is non-relativistic x>3

particle number per particle number per entropyentropy

Boltzmann equation for Boltzmann equation for YY

2 – 2 - scattering2 – 2 - scattering

as dominant annihilationas dominant annihilationand creation processand creation process

particle X in thermal equilibriumparticle X in thermal equilibrium( classical approximation )( classical approximation )

energy conservationenergy conservation

detailed balancedetailed balance( also for large T )( also for large T )

annihilation cross annihilation cross sectionsection

Boltzmann equation in Boltzmann equation in terms of terms of

cross sectioncross section

..

particle number per particle number per entropy entropy

in equilibriumin equilibrium

non –non –relativisticrelativistic

relativisticrelativistic

annihilation rateannihilation rate

freeze outfreeze out when annihilation rate when annihilation rate drops below expansion ratedrops below expansion rate

hot and cold relicshot and cold relics

hot relics hot relics freeze out when they freeze out when they are relativisticare relativistic

hot dark matter , hot dark matter , neutrinosneutrinos

cold relics cold relics freeze out when they freeze out when they are non-relativisticare non-relativistic

cold dark mattercold dark matter

hot relicshot relics

xxf f : freeze out inverse temperature: freeze out inverse temperature

YYEQ EQ is constant , approach to fixed point !is constant , approach to fixed point !

approach to fixed pointapproach to fixed point

Y > YY > YEQEQ : Y decreases towards Y : Y decreases towards YEQEQ . .Y < YY < YEQEQ : Y increases towards Y : Y increases towards YEQEQ . .

hot relicshot relics

for hot relics :for hot relics :

Y does not change , Y does not change ,

independently of detailsindependently of details

robust predictionsrobust predictions

neutrinosneutrinos

neutrino background radiationneutrino background radiation

ΩΩνν = = ΣΣmmνν / ( 91.5 eV h/ ( 91.5 eV h22 ) )

ΣΣmmνν present sum of neutrino massespresent sum of neutrino massesmmνν ≈ a few eV or smaller ≈ a few eV or smaller

comparison : electron mass = 511 003 comparison : electron mass = 511 003 eVeV

proton mass = 938 279 600 eVproton mass = 938 279 600 eV

cold relicscold relics

n=0 for s-wave scattering n=0 for s-wave scattering ( n=1 p-wave )( n=1 p-wave )

xxf f > 3> 3

s ~ Ts ~ T3 3 ~ x~ x-3-3

approximate solution approximate solution of Boltzmann equationof Boltzmann equation

early time x < 3early time x < 3

late time solutionlate time solution

YEQ strongly YEQ strongly Boltzmann suppressedBoltzmann suppressed

stopping stopping solutionsolution

freeze out temperaturefreeze out temperature

xxf f set by matching of set by matching of

early and late time early and late time solutionssolutions

ΔΔ ≈ ≈ ≈ ≈ YYEQ EQ ( x( xff ) )ff

cold relic abundancecold relic abundance

cold relic fractioncold relic fraction

cold relic fractioncold relic fraction

not necessarily order one !not necessarily order one !

cold relic abundance ,cold relic abundance ,mass and cross sectionmass and cross section

mm(m,(m,σσ

AA))

inversely proportional to cross section ,inversely proportional to cross section , plus logarithmic dependence of xplus logarithmic dependence of xff

baryon relics in baryon relics in baryon symmetric Universebaryon symmetric Universe

observed : Y ≈ 10 observed : Y ≈ 10 -10-10

anti-baryons in anti-baryons in baryon asymmetric baryon asymmetric

UniverseUniverse

relic WIMPSrelic WIMPS

only narrow range in m vs. only narrow range in m vs. σσ gives gives acceptable dark matter abundance !acceptable dark matter abundance !

relic WIMP fractionrelic WIMP fraction

strong dependence on mass and cross sectionstrong dependence on mass and cross section

Wimp boundsWimp bounds

relic stable heavy relic stable heavy neutrinosneutrinos

stable neutrinos with mass > 4-5 GeV allowedstable neutrinos with mass > 4-5 GeV allowed

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