Transcript Document
NEUTRINO DECOUPLE as Hot DM • Neutrinos are kept in thermal equilibrium by the creating electron pairs and scattering (weak interaction): e e e e • This interaction freezes out when the temperature drops to kT~MeV~ rest mass electrons – Because very few electrons are around afterwards • Argue that Neutrinos have Relativistic speeds while freezing out – kT >> rest mass of neutrinos(~eV) – They are called Hot Dark Matter (HDM) – Move without scattering by electrons after 1 sec. e.g., Neutrons • Before 1 s, lots of neutrinos and electrons keep the abundance of protons about equal to that of neutrons through – n + p + e• After 1 s free-moving neutrons start to decay. – n p + e- + – Argue that presently fewer neutrons in nuclei than protons thermal equilibrium number density • The thermal equilibrium background number density of particles is given by: 4 3 d p g 3 n 3 h 0 expE / kT 1 + for Fermions - for Bosons E m 2c 4 p 2c 2 • Where we have to change to momentum space and g is the degeneracy factor. E c 2 p 2 (m c2 ) 2 cp relativistic cp mc2 2 1 p m c2 2 m non relativistic cp mc2 • As kT cools, particles go from • From Ultrarelativistic limit. (kT>>mc2) particles behave as if they were massless 2 kT 4 g y dy n 3 y c ( 2 ) e 1 0 • To Non relativistic limit (kT<< 0.1mc2.) Here we can neglect the 1 in the occupancy number 3 ne mc 2 kT 4g y2 2 (2mkT ) e y dy 3 (2) 0 3 2 Number density of particles (annihilating/creating in a photon bath) Number Density N = g h 3 . o 4 p 3 3 d E e kT E c 2 p 2 m 2c 4 1 0.122 boson 3 KT g. hc g. (Relativistic) 3 2 2 mc 2 2KT .mc e KT 3 3 g 2 ,e, p g 1 h .c 0.091 fermion e Non-Relativistic Particles Freeze Out • Freeze-out of equilibrium (relativistic or non-relativistic) at certain temperature depending on number density, and crosssection. • Generally a particle A undergoes the reaction: A A • When the reverse reaction rate is slower than Hubble expansion rate, it undergoes freezeout. NA N ph A LOWweak interaction early freeze out while relativistic Freeze out A HIGHstrong interaction later freeze out at lower T m c2 kT A general history of a massive particle • Initially relativistic, dense (comparable to photon number density), – has frequent collisions with other species to be in thermal equilibrium and cools with CBR photon bath. Freeze-Out • Later, Relics Freeze-out of the cooling heat bath because – – – interactions too slow due to lower and lower density in expanding universe. This defines a “last scattering surface” where optical depth drops below unity. The number density falls with expanding volume of universe, but Ratio to photons kept constant. Number density of non-relativistic particles to relativistic photons • Reduction factor ~ exp(-mc2/kT), which drop sharply with cooler temperature. • Non-relativistic particles (relic) become rarer as universe cools (if maintain coupledequilibrium). smallest Collision cross-section • neutrinos (Hot DM) decouple from electrons (via weak interaction) while still relativistic kT>Δmc2. • Small Collision cross-section • Decouple at non-relativistic once kT<Δmc2 . Number density ratio to photon drops steeply with cooling exp(- Δmc2/kT). – anti-protons and wimps (Cold DM) etc. decouple (stop creating/annihilating) while non-relativistic. Abundant (CDM). – non-relativistic and combine into lower energy state. n H D He, e Neutral H. Neutrons/electrons Rarer than Hydrogen. • Tc~109K NUCLEOSYNTHESIS (100s) • Tc~5000K RECOMBINATION (106 years) (Redshift=1000) A worked-out exercise A A z Show at last scattering surface Optical depth = v n ph (z) 0 d (1+z)-n/2 ~ v (1+z) dz 0 dz ~ v (1+z)3-n/2 ~ vT 3-n/2 ~ 1 . z 3 where n=4 for radiation era. mc 2 Given that Freeze-out fraction ~ exp( ) kT and assume decouple at kT~mc2 /ln(1/ ), Argue cosmic abundance ~ m ~ T -1m /( v) ~ ( v) 1 dt dz dz A summary: Evolution of Number Densities of , P, e, PP P ee vv Num Density N Nο P R Rο H e H+ Now 1012 10 9 103 R Rο 3 109 Energetic Tail of Photon Bath dN dh 9 10 KTc hvc 25KTc hvc N hc KTc. hvc e .O 1 N ph KTc 2 e 25 252 10-9 "Freeze-out" KTc hvc 25 hv NB N ph # hardest photons ~ # baryons “Freeze-out”