11. Thermal History of the Universe
Radiation dominated approximation:
 c 2  S 4  S 3 S 1
no. particles  energy/particle
hence
k BT  S 1  t 1/ 2
where the connection to time t comes from the Freedman solution
S  t1/ 2 .
The above applies with any ‘hot’ matter, defined by
k BT mc 2
such that it is easy to excite PP pairs and that particle energies are
typically relativistic.
For any given species, once kBT falls below mc 2 the equilibrium
concentration rapidly becomes small,  e

mc2
k BT
.
Energy reference :
Proton: mpc2
1 GeV  109 1.6 1019 J
Electron: mec 2
0.5 MeV
Also quote temperatures in terms of the corresponding thermal
energy kBT .
(Figure)
Key annihilation energies
100 GeV  mc 2 for W  , Z bosons.
Above this energy (and corresponding temperature), weak and
electromagnetic interactions are largely equivalent.
1 GeV  mc2 for n, n , p, p
Above this nn , pp pairs readily excited.
Below 1 GeV we have only the residual net imbalance of
bayons n  p , negligible antibaryons.
The remaining imbalance was relatively tiny, is not understood, and
survives as the baryon abundance today. Baryon number n  p is
conserved [except perhaps in Black Holes, and GUT suggests proton
might decay in 1035 years].
0.5 MeV  mc 2 for e 
Above this, plentiful electron-positron pair excitation.
Below 0.5 MeV only the residual e  balancing charge of residual
protons. Preserved by charge conservation.
13.6 eV H   e  H deionisation (‘recombination’) of Hydrogen
eliminates the last free charges.
12. Fate of Residual Baryons: Nucleosynthesis
Background

Baryonic matter today is roughly 22-24%
and almost all the remainder is 1 H .

Corresponding neutron ratio is

Stars have not had long enough to produce much of this
Helium

Most is explained by Big Bang origins.
4
He by mass,
n
 1/ 7
p
Key Steps


kBT  1 GeV residual net baryon number n  p not explained
Equilibrium reactions
0.8MeV  p  e 
n  e
1.8MeV  p   e
n  e
where  e ,( e )  electron (anti) neutrino.
Provided e  , e  , e , e all in equilibrium, these


 n p
13.6 eV
n
k BT
k BT
e
e
p
k BT  2.3MeV weak interactions too slow to keep  e , e in
equilibrium. Beyond this point the neutrinos are decoupled,
concentration diluting as c  S 3 , begging to be observed.
Electrons stay in equilibrium due to faster e/m reaction
e  e
2
E E
imply




Due to higher cross-sections, baryon-neutrino reactions do
not shut off until about 0.8MeV , leaving
n
13.6 eV
0.8 MeV
e
 1/ 5 ,
 p
 lepton
when the Universe is about 20s old.
Neutrons can also decay spontaneously
n  p  e   e
with a lifetime   15 minutes , so cosmology has just that
long to save its neutrons!
Key neutron preservation step is formation of deuterons
d  2 H  by
n p
d    2 MeV
which is highly exothermic but still impeded by the back
reaction due to low baryon and high photon concentrations.
This only runs net forwards around k BT  0.1MeV ,
fortunately only t  2.5 minutes on the cosmic clock, leaving
us
n
n
 et /  
 1/ 7
p
 p lepton
where these remaining neutrons are safely locked up in stable
deuterons.
I have desperately simplified a complicated story above, but the
bottom line is excellent agreement between Big Bang
nucleosynthesis and the observed abundance of Helium in the
universe.

From Deuterium to Helium can take its time, first reaching
the two nuclei of mass 3 by every conceivable route:
d  p  3 He 2  
d  n  t ,
t  3H 
d  d  3 He 2  n
d d t p
Regular mass 4 Helium then follows by
3
He2  n  4 He2



t  p  4 He2
There are no nuclei of mass 5 which constitutes a major
bottleneck.
Mass 6 offers only 6 Li which is still unstable but can live
long enough that there is some yield of
6
Li  n  7 Li
Mass 8 being another bottleneck, production of heavier
nuclei is negligible.
Below kBT  0.1MeV little changes for a long time, with
p, 4 He, 3He, e interacting via photons, which are heavily scattered.
We could only ‘see’ events or structures from this era by their
perturbation to the neutrino background, or perhaps gravitational
waves.
13. Deionisation of Hydrogen and the CMBR
pe
H  2 with 13.6 eV available to the photons, but the low
concentration of p  e holds up the forward reaction until
kBT  0.3eV or T  4000K .
This is of the same order as the visible surface of the sun, for related
reasons.
Key idea: Last Scattering Surface where photons had last
interaction with matter. This is what we ‘see’ when we image the
microwave background. It is equivalent to imaging the surface of
the sun, but redshifted by z  1100 to
T  2.728K (Cobe, 19__)
(most accurate Black Body spectrum ever observed in any branch of
science).
The equivalent neutrino background did not benefit from the
deionisation energy of H, so
T  1.949 K
remains predicted awaiting observation.
Structure in CMBR
Cobe saw a dipolar perturbation  T T  103 , interpreted as our
peculiar motion
v  365 km / s
relative to the global Hubble flow (and CMBR).
When this is subtracted, the remaining intrinsic structure has
 T T  105 which it is an even more recent triumph to have
mapped out in detail.
Angular Correlation Function
 T    T    
C ( ) 
T
T


1
 (  1) C P (cos )
4 2
 angular wavevector ; angular lengthscale  60

.
 200    0.3 :
Sakharov oscillations, acoustic standing waves inside LSS:
higher peaks predicted and now observed.
Peak at
Particle Horizon of LSS
The particle horizon is how far away a past event could have
occurred and still have influence on the event of interest. Looking
back from the LSS along a light ray c 2 dt 2  S (t )2 d 2  0 (and
ignoring curvature) gives this distance in terms of
t1
ct
cdt
1  
2 1
S (t )
S1
0
where the estimate of the integral assumes radiation dominance up to
the LSS, S (t )  t1/ 2 .
Points on the LSS further apart than 2 1S1 can have no past event in
common back to the beginning of time, setting a severe limit on the
range of intrinsic correlations in the CMBR.
We can relate distances in the LSS (e.g.  1S1 ) to small angles in the
observed microwave sky by
 1S1   0 S11
where  0 is the comoving distance between us and the LSS.
Using a light ray from the LSS to us, we obtain
t0
0  
t1
ct
cdt
3 0
S (t )
S0
using cold matter dominance, S (t )  t 2/ 3 , and noting that the
contribution form the lower limit is comparatively negligible.
Combining these results gives us
1/ 2

2t / S
2 S 
1  1  1 1   1   0.02 radians
 0 3t0 / S0 3  S0 
where we have used S (t )  t 2/ 3 and the observed redshift of the
CMBR.
Converted to degrees, this gives the maximum range of common
influence about 21  3 , and such an angular scale can indeed be
seen in the CMBR correlations.
Reheat
The WMAP satellite data make quite clear the existence of a weaker
feature in the CMBR angular correlation, extending to of order 90 .
This is confidently attributed to scattering of CMBR by subsequent
hydrogen clouds reheating as density fluctuations grow, leading of
course to star and galaxy formation. These are much more recent
events, so at much smaller  0 and are seen to influence larger angles
in the sky.
14. Summary of Big Bang Relics
Here are the key relics of the Big Bang.
Residual net baryons p  n since kBT  1GeV
-observed but not understood
Neutrino background
-predicted
decoupled since kBT  2.3MeV
Neutrons stabilised since kBT  0.1MeV (2minutes) as d  He
- observed
CMBR photons from kBT  0.3eV
- observed including angular structure
Reheat from primordial H cloud collapse
- observed
In a loose sense you can think of them as signatures of the strong,
weak, electromagnetic and gravitational interactions each
dominating events in turn.
Remaining Puzzles
However convincing the agreement of cosmological observations
might be, it remains a concern that the Universe today appears quite
recently to have turned from deceleration under the influence of
matter to acceleration under the influence of the cosmological
constant.
Most people can accept the Universe being predominantly
propulated by dark matter, which we may not be able to see but
which we can certainly see the gravitational influence of in the likes
of galaxy rotation curves. Predominant dark energy is for some a
step too far.
Many things are beyond the scope of this course, including the
arguments about structure in the distribution of galaxies. At every
turn in the story there has been too much. Recently there has been
some matching up of the galaxy distribution with the underlying
structure in the CMBR, but in the last two months it has been
asserted that the observed galaxy clustering requires  m  1 to be
able to develop. If that claim holds up, this is the most serious
challenge to the new ‘standard cosmology’.