Topic 3

Primordial nucleosynthesis

Evidence for the Big Bang !  Back in the 1920s it was

generally thought that the Universe was infinite

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

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

recessional velocity (and hence red shift) 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)/Mpc and 1/H0 = Age of Universe

Olbers’ paradox !  Steady state Universe is:

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

!  Therefore 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 the assumption that the Universe is infinite must be flawed

Radio sources

!  Based on observations of radio sources of different strengths (so-called 2C and 3C surveys)

!  The number of radio sources versus source strength concludes that the Universe has evolved from a denser place in the past

!  This again appears to rule out the so-called Steady State Universe and gives support for the Big Bang Theory

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.

!  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

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 reasonable agreement 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 these calculations

capture cross-sections measured at Los Alamos

during World War II were used

(1 MeV neutrons =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) x (the volume) x (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 x beam area

Integration yields N = N0 exp(- nσx)

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

αβγ theory - success and failure !  Abundance for He agrees well with observation !  By 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

!  Results carved the way for calculations of thermonuclear fusion

!  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 then annilihilation rate > creation rate

Big Bang; Underlying Principles II

!  Normal physics laws (including standard model of particle physics)

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

homeogeneous and isotropic) Robertson-Walker metric

!  Expansion of the Universe is governed by field equations of GR

The Big Bang

Time

Space

Key events after Big Bang Time 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 Bang Time Temp/Energy Event 101 s T = 1010 K Neutrinos “decouple” from the

cosmic plasma (cross-section falls dramatically)

102 s T = 7.5-6x109 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 atoms Neutron decay continues, n:p ratio changing

Next 103 s

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

Key events after Big Bang Time Temp/Energy Event

~ 103 s to 400,000 years

T ~ 108 or 9 K to T = 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 = 3000 K Photons can no longer ionize, photons decouple, “last scattering surface”. Origin of CMBR.

Fundamental forces

Cosmic Microwave Background

Cosmic Microwave Background

Very close to a perfect thermal (Black Body) spectrum with a temperature

of 2.7K

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- (+0.8MeV) � n + νe

(iii) n � p + e- + νe (+ 0.8 MeV) !  Note that reaction (ii) 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 = Δm, protons and neutrons are nearly equal in

number •  When Δm becomes significant compared to kT, the neutron-proton

ratio is given by the Boltzmann factor exp(−Δmc2/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

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

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Primordial nucleosynthesis !  At this point kT is too high

for primordial nucleosynthesis to start (formation of nuclei) due to dissociation

!  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 + n 2H + 2H ⇔ 3H + p

3H + 2H ⇔ 4He + n 3He + 2H ⇔ 4He + p

2H + 2H ⇔ 4He 3He + 4He ⇔ 7Be + γ

3H + 4He ⇔ 7Li + γ 7Be + n ⇔ 7Li + p

7Li + p ⇔ 24He Note: ions not atoms

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

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Abundances vs time Note that a

neutron:proton ratio of 0.135:1

is equivalent to 12:88

Assuming that the 12 neutrons go to forming

4He we would expect

76% Hydrogen (1H) and

24% Helium (4He) -  in excellent agreement

with observation

Modern day abundances !  Comparison of modern

day elemental abundances from primordial nucleosynthesis 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

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