Evolution of the Universe, Particle Acceleration, and Cosmic Rays - PowerPoint PPT Presentation

Evolution of the Universe, Particle Acceleration, and Cosmic Rays Particle Astrophysics Exercise Session #1 Erik Strahler 25/03/11

Plan • Review important concepts • Introduce new material • Work through some examples • Assign work to be done at home – First assignment due May 6th 25/03/11 Erik Strahler - Particle Astrophysics 2

Overview • Evolution of the Universe – The FRW Metric – The decoupled radiation • Cosmic Accelerators – Interactions and reaction products • Cosmic Rays – Propagation and energy loss – Practical Methods: Monte Carlo Simulation 25/03/11 Erik Strahler - Particle Astrophysics 3

Describing the Universe • Einstein equation: • +FLRW metric: • Friedmann equation: 2 & ⎛ ⎞ π ρ 2 8 R G kc ⎜ ⎟ = = − 2 H ⎜ ⎟ 2 3 ⎝ ⎠ R R = ⋅ ( ) ( ) D t r R t 25/03/11 Erik Strahler - Particle Astrophysics 4

Maximum size • For a closed universe, what is the maximum size? 2 ⎛ & ⎞ π ρ 2 8 R G kc ⎜ ⎟ = = − 2 H ⎜ ⎟ 2 3 ⎝ ⎠ R R & = + = 1 , 0 k R 25/03/11 Erik Strahler - Particle Astrophysics 5

Maximum size • For a closed universe, what is the maximum size? 2 ⎛ & ⎞ π ρ 2 8 R G kc ⎜ ⎟ = = − 2 H ⎜ ⎟ 2 3 ⎝ ⎠ R R & = + = 1 , 0 k R 2 3 c 2 ⇒ = R max π ρ 8 G 3 π ρ 4 R = max M 3 2 GM ⇒ = R max 2 c 25/03/11 Erik Strahler - Particle Astrophysics 6

Exercise • Find the total time to the big crunch assuming a closed universe (k=1) and total M=10 23 M sun (M sun = 2x10 30 kg). • You will need to make the substitution 2 GM − = θ 2 2 2 tan c c R • Show your work! 25/03/11 Erik Strahler - Particle Astrophysics 7

CMB decoupling 25/03/11 Erik Strahler - Particle Astrophysics 8

The Cooling Universe • Expansion leads to dropping temperature (density) • In general, particles are freely produced until kT drops below their mass – ~100 GeV: Quark-gluon Plasma – ~200 MeV-10MeV: Hadronization • Most hadrons decay, leading to lots of e, p, γ , ν which freely interact – ~3 MeV: electron density is decreasing • Neutrino freeze out • BBN 25/03/11 Erik Strahler - Particle Astrophysics 9

The Cooling Universe: 2 – ~13.6 eV: formation of neutral Hydrogen – ~.25 eV: sufficiently small electron density to stop reactions, and decouple photons • Leads to CMB • Start of matter dominated universe 25/03/11 Erik Strahler - Particle Astrophysics 10

Evolution of Radiation • At t~380,000s, photons are decoupled from matter and evolve independently. • Photons obey Bose-Einstein statistics and thus have intensity given by Planck’s law for a black- body: ν 3 2 1 h ν = ( , ) I T ν 2 h c − 1 kT e 25/03/11 Erik Strahler - Particle Astrophysics 11

Evolution of Radiation • We can also write this in terms of the spectral energy density, in units of the total energy / unit volume / unit frequency π π ν 3 4 8 1 h ν = ν = ( , ) ( , ) u T I T ν 3 h c c − 1 kT e 25/03/11 Erik Strahler - Particle Astrophysics 12

Exercises • Using the previous information, determine the relationship between the energy density of radiation and the temperature of the expanding universe. • Use the above relationship to find a function relating the photon number density to the temperature and use this to find the current value. Compare to the value shown in lecture. 25/03/11 Erik Strahler - Particle Astrophysics 13

Cosmic Accelerators • Variety of Sources – Supernovae – Pulsars – AGN – GRBs – … 25/03/11 Erik Strahler - Particle Astrophysics 14

General Mechanism • Some central engine accelerates protons, typically via repeated crossings of variable magnetic fields in shock fronts. – Produces a spectrum following a power law of ~E -2 • Protons interacts with each other, and with ambient or co-accelerated photons, electrons + γ → • Creates high energy photons, p X neutrinos + → p p X 25/03/11 Erik Strahler - Particle Astrophysics 15

Relativistic Kinematics • 4-vectors which obey Lorentz transformations have products that are invariants. = ( , , , ) ds ct x y z ⇒ = − − − 2 2 2 2 2 ( ) ds ct x y z E = ( , , , ) P p p p x y z c E 2 ⇒ = − 2 2 ( ) P p c + 2 , ( ) also PQ P Q 25/03/11 Erik Strahler - Particle Astrophysics 16

Example • Calculate the minimum projectile energy in the rest frame of the target for a pp interaction that produces π 0 (taking into account baryon and charge conservation!). – m π = 135 MeV – m p = 938.3 MeV 25/03/11 Erik Strahler - Particle Astrophysics 17

Exercises • Calculate the threshold photon energy in the → + γ π + proton rest frame for the interaction , p X where the π + mass is 139.6 MeV. Infer what X can be. + + π → μ + ν • In the subsequent decay , calculate μ what fraction of the pion energy the neutrino takes (in the rest frame of the pion). m μ =105.7 MeV 25/03/11 Erik Strahler - Particle Astrophysics 18

Cosmic Rays at Earth • Properties • Lifetime • Energy loss 25/03/11 Erik Strahler - Particle Astrophysics 19

Energy Spectrum • Primary CRs: – 86% protons – 11% Helium ions – 1% heavy ions – 2% electrons • Composition depends on energy • Produced in astrophysical sources 25/03/11 Erik Strahler - Particle Astrophysics 20

Conceptual Question • Most primary cosmic rays that interact in the atmosphere are protons or heavy ions. Only about 2% are electrons. Should we therefore expect that a net positive charge exists on the Earth due to CR bombardment? 25/03/11 Erik Strahler - Particle Astrophysics 21

Atmospheric Muons • Example: What is the minimum energy required for a cosmic-ray induced muon to reach the surface of the Earth (sea level) if it is produced at a height of 20 km? – τ μ = 2.2 μ s – m μ = 105.7 MeV 25/03/11 Erik Strahler - Particle Astrophysics 22

Atmospheric Muons • Example: What is the minimum energy required for a cosmic-ray induced muon to reach the surface of the Earth (sea level) if it is produced at a height of 20 km? – τ μ = 2.2 μ s – m μ = 105.7 MeV 25/03/11 Erik Strahler - Particle Astrophysics 23

Energy Losses • Ionization and atomic excitation: interactions with electrons in the media – continuous process [mip: particles at the minimum of ionization 2 MeV/g/cm 2 ] • Radiative: discrete process and stochastic – Bremmsstrahlung: radiation emitted by an accelerated or decelerated particle through the field of an atomic nuclei – Pair production: μ +N → e+e- – Photonuclear : inelastic interaction of muons with nuclei, produces hadronic showers 25/03/11 Erik Strahler - Particle Astrophysics 24

Energy Losses • dE/dx = a(E)+b(E) E – Ionization + stochastic losses (dominate above 1 TeV) 25/03/11 Erik Strahler - Particle Astrophysics 25

Example: Energy Losses • What is the range of a 100 GeV muon in rock? – ρ rock = 2.65 g / cm 3 – a = 2 MeV / g / cm 2 – b = 4.4x10 -6 cm 2 / g 25/03/11 Erik Strahler - Particle Astrophysics 26

Example: Energy Losses • What is the range of a 100 GeV muon in rock? – ρ rock = 2.65 g / cm 3 – a = 2 MeV / g / cm 2 – b = 4.4x10 -6 cm 2 / g 25/03/11 Erik Strahler - Particle Astrophysics 27

Exercise • Derive a general equation giving the remaining energy of a muon after traversing a slant depth X if it had initial energy E 0 . Assume average energy losses apply. Describe the behavior of the resulting function for large and small depths and identify the transition point. 25/03/11 Erik Strahler - Particle Astrophysics 28

Practical Methods: Monte Carlo • Model a situation by randomly sampling from a known (or approximated) underlying distribution – Useful when exact, analytical results are too difficult to achieve – Also useful for achieving a statistical result 25/03/11 Erik Strahler - Particle Astrophysics 29

MC: How to Achieve? • First, we need to be able to uniformly generate statistically independent values (typically in a range [0,1] ) • Next, map a probability distribution of our variable of interest into a parameter that can be sampled. i.e. How do we determine random values of x when we know f(x) ? 25/03/11 Erik Strahler - Particle Astrophysics 30

Inverse Transform Method − ∞ < < ∞ • If we have a PDF of form f(x) defined on x (normalized to 1), then its Cumalitive Distribution Function F(a) expresses the probability that x<a and is given by a ∫ = ( ) ( ) F a f x dx − ∞ • Now, U=F(X) is a random variable that occurs on the interval [0,1]. We can generate random values from the CDF by finding X=F -1 (U) 25/03/11 Erik Strahler - Particle Astrophysics 31

Inverse Transform Method Example ≤ ≤ ⎛ ⎞ 2 0 1 x x = ⎜ ⎟ ( ) f x ⎜ ⎟ 0 ⎝ ⎠ otherwise < ⎛ ⎞ 0 0 x ⎜ ⎟ ⎜ 1 ⎟ ∫ = = ≤ ≤ 2 ( ) 2 0 1 ⎜ ⎟ F x xdx x x ⎜ ⎟ 0 ⎜ ⎟ > 1 1 ⎝ ⎠ x = 2 U X − 1 = = 1 / 2 ( ) X F U U x 25/03/11 Erik Strahler - Particle Astrophysics 32