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Topic 3

Primordial nucleosynthesis

Evidence for the Big Bang

Back in the 1920s it was generally thought that the Universe was static

However a number of experimental observations started to question this, namely:• Red shift and Hubble’s Law• Olbers’ Paradox• Radio sources• Existence of CMBR

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Red shift and Hubble’s Law We have already discussed red shift in the

context of spectral lines (Topic 2) Crucially Hubble discovered that the red shift

(and hence recessional velocity) of galaxies increases linearly with their distance from us according to the famous Hubble Law

V = H0d where

H0 = 69.3 ± 0.8 (km/s)/Mpcand 1/H0 ~ Age of Universe

Olbers’ paradox If we assume Universe is

infinite in space and time isotropic (sky looks the same in all directions), homogeneous (our location in the Universe isn’t special) and not expanding

Then an observer choosing to look in any direction should eventually see a star

This would lead to a night sky that is uniformly bright (as a star’s surface)

This is not the case and so at least one of the assumptions must be flawed

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Aside: Steady State Universe

Expanding but infinitely old, as new matter is continuously created at low rate to maintain constant density

Explains Hubble’s law and resolves Olbers’ Paradox, as redshift lowers apparent temperature of distant objects

But not consistent with observation that appearance of universe changes with redshift(e.g. quasars and radiogalaxies more common at high redshift) or with CMB

Cosmic Microwave Background CMBR was predicted as early as 1949 by Alpher and

Herman (Gamow group) as a “remnant heat” left over from the very hot and dense initial Universe

They predicted that after the Big Bang the Universe should “glow” in the gamma ray part of the spectrum

This will subsequently cool as the Universe expands shifting the wavelength of this “last light” to a temperature of ~5K

Eventually observed in 1965 by Penzias and Wilson

The CMBR is now a very powerful tool for cosmologists

Recent experiments such as COBE and WMAP have measured the CMBR anisotropies at the 10-5 level

Gives us information on Big Bang, Dark Matter, etc.

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Subsequently they proposed a single process for all elemental abundances in the Universe − that of neutron capture

Protons via β-decay: n p + e- + νe

First step: p + n 2H + γ

αβγ theory (Origin of Chemical Elements)

Actually Alpher & Gamow: Bethe included (by Gamow) as a joke

Proposed an early Universe that was hot and dense

Assumed that the Early Universe consisted only of neutrons

As the temperature fell neutron decay to protons was possible

αβγ theory

νe

νe

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αβγ theory - abundances Successive neutron

capture creates heavier elements

At each step the progress controlled by the balance between the rate of production and the rate of destruction

By setting up and solving a sequence of differential equations of this type, a distribution could be produced in reasonableagreement with the trend of the observed abundances

dNA/dt = F(S,T)[σ A-1NA-1 - σANA]F is collision frequency (function of thermodynamic state variables) NA is the no. of atoms with atomic no. A

σA is the neutron capture cross-section

For thesecalculations

capturecross-sectionsmeasured at Los Alamos

duringWorld War IIwere used

(1 MeVneutrons=1010K)

Cross-sections (quick revision) Consider the simple case in which a

beam of particles is incident on nuclei of some type, then the cross-section is the probability of a particular process occurring per target nucleus, per incident particle

The total area “blocked out” is the number of nuclei per unit volume times the volume times σ. Thus the fraction of the beam which is removed by the reaction is

In neutron capture the rate at which the reaction is occurring depends upon the relative velocity v of the particles and target nuclei and is given by the product of particle density, the relative velocity, the cross section and the total number of target nuclei.

We shall discuss neutron capture further in understanding the production of elements heavier than Iron

dN/N = −nσ dxwhere n = number density × beam area

Integration yields N = N0 exp(−nσx)

or N = N0 exp(−x/λ )where λ is the mean free path

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αβγ theory - success and failure Abundance for He agrees well with observation Splitting the elements into 15 “groups” by atomic weight and

using an average cross-section for each group gives a reasonable fit to abundance data

BUT predicted abundances for heavier elements were incorrect

Problem getting past A = 4 due to lack of stable elements with A = 5, 8

Also, heavy element abundances in stars depend on age (older less)

Discussion is relevant to neutron capture topic later

This is an extract from the “Chart of nuclides”

Big Bang: Underlying principles I

Universe expanded some 14 billion years ago from a singularity

At extremely high temperatures elementary particles can simply be created from thermal energy kT = mc2 (essentially E = mc2)

After the BB the Universe expands and cools As temperatures fall below the threshold

temperature for particle production thenannihilation rate > creation rate

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Big Bang; Underlying Principles II

Normal physics laws (including standard model of particle physics)

Small matter-antimatter asymmetry Gravitation described by General Relativity Cosmological principle (Universe is

homogeneous and isotropic) Robertson-Walker metric

Expansion of the Universe is governed by field equations of GR

The Big Bang

Time

Space

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Key events after Big BangTime Temp/Energy Event

10-43 s kT = 1019 eV Planck era, quantum gravity, prior to this all forces one, gravity first to decouple, many exotic particles

10-35 s kT = 1015 eV Inflation starts, Strong nuclear force decouples

10-10 s -10-4 s

T = 1015 K − 1012 K

Free electrons, quarks, photons, neutrinos all strongly interacting

10-4 s -101 s

T = 1012 K − 1010 K

Free electrons, protons, neutrons, photons, neutrinos all strongly interacting

Key events after Big BangTime Temp/Energy Event

101 s T = 1010 K Neutrinos “decouple” from the cosmic plasma (cross-section falls dramatically)

102 s T=(7.5)6×109 K Pair production of e+e- ceases

102 s kT = 0.8 MeV Proton:neutron ratio is frozen

Next 300 s

Thermal energy still high enough to photodissociate atomsNeutron decay continues, n:p ratio changing

Next 103 s

Primordial nucleosynthesis starts Note ions not atoms due to mean thermal energy

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Key events after Big BangTime Temp/Energy Event

~ 103 s to 400,000years

T ~ 108 or 9 K toT = 3000 K

“Dark ages”: Universe is a sea of free nuclei, electrons and photons. Photons Thomson scatter off electrons so Universe remains opaque to photons. Physics in this period is less well-established.

380,000 years

T = 3000K Photons can no longer ionize, photons decouple, “last scattering surface”. Origin of CMBR.

Fundamental forces

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Cosmic Microwave Background

Cosmic Microwave Background

Very close to aperfect thermal(Black Body)

spectrum with a temperature

of 2.7K

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The neutron:proton ratio

The main 3 reactions involved in determining the number of protons and neutrons in the early Universe are:

(i) n + e+ p + νe (+1.8 MeV) (ii) p + e- (+ 0.8 MeV) n + νe

(iii) n p + e- + νe (+ 0.8 MeV) Note that reaction 2 is endothermic in a left-

right direction i.e. requires energy into the system (KE of incoming particles) in order to proceed

The neutron:proton ratio At T > 1010 K, kT > 1 MeV, t < 1 s, reactions (i) and (ii)

maintain protons and neutrons in thermal equilibrium• When kT >> mn – mp = ∆m, protons and neutrons are nearly equal in

number

• When ∆m becomes significant compared to kT, the neutron-proton ratio is given by the Boltzmann factor exp(−∆m/kT)

At T ~ 1010 K, kT ~ 0.8 MeV, t ~ 1 s, the reaction rates for (i) and (ii) become slow compared to the expansion rate of the universe• neutrinos decouple (weak interaction rate slow compared to

expansion rate)

• e+e− pair creation suppressed (γ energies drop below 0.511 MeV)

• neutron:proton ratio “freezes out”

Below this temperature only reaction (iii) continues

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The neutron:proton ratio We use the Boltzmann distribution to estimate the

n:p ratio at this point

hence

where kT = 0.8 MeV and (mn − mp) = 1.3 MeV/c2

This yields a value of Nn:Np ~ 0.2

N ∝ m3

2 exp − mc2

kBT

Nn

N p

= mn

mp

32

exp −(mn − mp )c 2

kBT

Primordial nucleosynthesis At this point kT is too high

for primordial nucleo-synthesis (formation of nuclei) to start due to photodissociation

Therefore reaction (iii) continues in the left-right direction – this is neutron decay

After a further 300 seconds primordial nucleosynthesis starts

p + n 2H + γ2H + 2H 3He + n2H + 2H 3H + p

3H + 2H 4He + n3He + 2H 4He + p

2H + 2H 4He3He + 4He 7Be + γ

3H + 4He 7Li + γ7Be + n 7Li + p

7Li + p 24HeNote: ions not atoms

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Solved problem If the neutron:proton ratio starts at 0.2 and the neutron continues to decay

for a further 300 seconds what is the neutron:proton ratio at the end of this period given that the neutron’s lifetime is 890 seconds?

The neutron’s lifetime is 890 seconds therefore in 300 seconds:

Therefore the fraction of neutrons that have decayed = 0.286 Next we write

where = 0.2 and d=0.286 to give = 0.135

N

N0

= exp − t

τ

= exp − 300

890

= 0.714

N n

N p

t= 300s

= N n (1− d)

N p + dN n

=

N n

N p

(1− d)

1+ dN n

N pN n

N p

N n

N p

t= 300s

Abundances vs timeNote that a

neutron:protonratio of0.135:1

is equivalent to 12:88

Assuming that the 12 neutrons

go to forming4He

we would expect 76% Hydrogen (1H)

and24% Helium (4He)

- in excellent agreement

with observation

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Modern day abundances

Comparison of modern day elemental abundances from primordial nucleo-synthesis can also give important cosmological information such as the baryon density or the baryon to photon ratio

Concordance with CMB is important check on theory

0.038 ≤ Ωb ≤ 0.048

Summary

Big Bang Nucleosynthesis (BBNS) successfully predicts the production of light elements shortly after the Big Bang

The thermal history of the early Universe and nuclear physics are used to explain the sequence of events

Light element abundances can be accurately predicted and related to cosmological parameters