Topic 3. Evidence for the Big Bang

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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 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

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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

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

V = H

₀

d where

H

₀

= 69.3 ±0.8 (km/s)/Mpc and 1/H

₀

= 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

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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.

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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^- + ν_e

!  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

ν_e

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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 reasonable agreement with the trend of the observed abundances

dN_A/dt = F(S,T)[σ _A-1N_A-1 - σ_AN_A]

F is collision frequency (function of thermodynamic state variables)

N_A 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 dx

where 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

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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

! 

As temperatures fall below the threshold temperature for particle production then

! 

Expansion of the Universe is governed by field equations of GR

The Big Bang

Time

Space

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Key events after Big Bang

Time Temp/Energy Event

10^-43s kT = 10¹⁹eV Planck era, quantum gravity, prior to this all forces one, gravity first to decouple, many exotic particles 10^-35s kT = 10¹⁵eV Inflation starts, Strong nuclear

force decouples 10^-10s

-10^-4s

T = 10¹⁵K – 10¹²K

Free electrons, quarks, photons, neutrinos all strongly interacting 10^-4s

-10¹s

T = 10¹²K – 10¹⁰K

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

Key events after Big Bang

Time Temp/Energy Event

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

10²s T = 7.5-6x10⁹K Pair production of e⁺e^- ceases 10²s 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 10

³

s

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

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Key events after Big Bang

Time Temp/Energy Event

~ 10³s to

400,000 years

T ~ 10^{8 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

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Cosmic Microwave Background

Very close to a perfect 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 + ν

_e

(+ 1.8 MeV) (ii) p + e

^-

(+0.8MeV) n + ν

_e

(iii) n p + e

^-

+ ν

_e

(+ 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 > 10¹⁰K, kT > 1 MeV, t < 1 s, reactions (i) and (ii) maintain protons and neutrons in thermal equilibrium

•  When kT >> m_n – m_p= 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)

_p

~ 0.2

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N ∝ m

³²

exp − mc

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_B

T

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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

! 

7

Li + p ⇔ 2

⁴

He

Note: 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

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N

N₀ = exp − t τ

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( ) = exp −300 890

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( ) = 0.714

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N_n N_p

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t= 300s

= N_n(1− d) N_p + dN_n =

N_n

N_p (1− d) 1+ d N_n

N_p

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N_n N_p

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N_n N_p

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t= 300s

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

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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

! 