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

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Plan

• Review important concepts
• Introduce new material
• Work through some examples
• Assign work to be done at home

– First assignment due May 6th

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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

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Describing the Universe

• Einstein equation:
• +FLRW metric:
• Friedmann equation:

) ( ) ( 3 8

2 2 2 2

t R r t D R kc G R R H ⋅ = − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ρ π &

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Maximum size

• For a closed universe, what is the maximum size?

, 1 3 8

2 2 2 2

= + = − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = R k R kc G R R H & & ρ π

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Maximum size

• For a closed universe, what is the maximum size?

, 1 3 8

2 2 2 2

= + = − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = R k R kc G R R H & & ρ π

2 max 3 max 2 2 max

2 3 4 8 3 c GM R R M G c R = ⇒ = = ⇒ ρ π ρ π

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Exercise

• Find the total time to the big crunch assuming a

closed universe (k=1) and total M=1023 Msun (Msun= 2x1030 kg).

• You will need to make the substitution
• Show your work!

θ

2 2 2

tan 2 c c R GM = −

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CMB decoupling

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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

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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

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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:

1 1 2 ) , (

2 3

− =

kT h

e c h T I

ν

ν ν

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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

1 1 8 ) , ( 4 ) , (

3 3

− = =

kT h

e c h T I c T u

ν

ν π ν π ν

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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.

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Cosmic Accelerators

• Variety of Sources

– Supernovae – Pulsars – AGN – GRBs – …

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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

• Creates high energy photons,

neutrinos

X p p X p → + → +γ

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Relativistic Kinematics

• 4-vectors which obey Lorentz transformations

have products that are invariants.

2 2 2 2 2 2 2 2 2

) ( , ) ( ) , , , ( ) ( ) , , , ( Q P PQ also c E P p p p c E P z y x ct ds z y x ct ds

z y x

+ − = ⇒ = − − − = ⇒ = p

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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!).

– mπ = 135 MeV – mp = 938.3 MeV

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Exercises

• Calculate the threshold photon energy in the

proton rest frame for the interaction , where the π+ mass is 139.6 MeV. Infer what X can be.

• In the subsequent decay , calculate

what fraction of the pion energy the neutrino takes (in the rest frame of the pion). mμ=105.7 MeV X p + → + π γ

μ

ν μ π + →

+ +

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Cosmic Rays at Earth

• Properties
• Lifetime
• Energy loss

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Energy Spectrum

• Primary CRs:

– 86% protons – 11% Helium ions – 1% heavy ions – 2% electrons

• Composition depends
• n energy
• Produced in

astrophysical sources

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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?

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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?

– τμ = 2.2 μs – mμ = 105.7 MeV

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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?

– τμ = 2.2 μs – mμ = 105.7 MeV

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Energy Losses

• Ionization and atomic excitation:

interactions with electrons in the media

– continuous process [mip: particles at the minimum of ionization 2 MeV/g/cm2]

• 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

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Energy Losses

• dE/dx = a(E)+b(E) E

– Ionization + stochastic losses (dominate above 1 TeV)

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Example: Energy Losses

• What is the range of a 100 GeV muon in rock?

– ρrock = 2.65 g / cm3 – a = 2 MeV / g / cm2 – b = 4.4x10-6 cm2 / g

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Example: Energy Losses

• What is the range of a 100 GeV muon in rock?

– ρrock = 2.65 g / cm3 – a = 2 MeV / g / cm2 – b = 4.4x10-6 cm2 / g

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Exercise

• Derive a general equation giving the remaining

energy of a muon after traversing a slant depth X if it had initial energy E0. Assume average energy losses apply. Describe the behavior of the resulting function for large and small depths and identify the transition point.

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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

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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)?

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Inverse Transform Method

• If we have a PDF of form f(x) defined on

(normalized to 1), then its Cumalitive Distribution Function F(a) expresses the probability that x<a and is given by

• 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)

∞ < < ∞ − x

∫

∞ −

=

a

dx x f a F ) ( ) (

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Inverse Transform Method

Example

2 / 1 1 2 1 2

) ( 1 1 1 2 ) ( 1 2 ) ( U U F X X U x x x xdx x x F

• therwise

x x x f

x

= = = ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ > ≤ ≤ = < = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ≤ ≤ =

−

∫

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Exercise

• We’ve seen that cosmic ray fluxes can be characterized

by power law spectra due to the acceleration mechanisms in their sources. Assuming a generation spectrum dN/dE=AE-γ defined in the energy interval [E1,E2], apply the Inverse Transform Method to find an analytical solution to generate events from this distribution. Write a short program (with any language you like) to generate and plot events with γ=2 and γ=1 for your choice of normalization constant and energy range.