So, we can take a measure of the 
temperature of the universe and we have 
really good evidence that the universe is 
isotropic and homogeneous. 
We'll see that we learn a lot more from 
the cosmic background in the 2000s than 
just that it exists but that's not 
evidence, what I've given you so far is 
far from convincing evidence for the Big 
Bang. 
Some of the best evidence for the Big 
Bang is a story that is rarely told and 
is, because it's a bit technical, but I 
want to tell it and I want I think we can 
appreciate what's going on. 
So, you know this is the story of Big 
Bang nucleosynthesis and it's interesting 
we talked about stars and nucleosynthesis 
and the R process and the S process and 
supernovae as creating all the heavy 
elements. 
But we never discussed the production of 
Helium, of course, Helium is produced in 
stars. 
But we saw that not in a quantity that 
would significantly alter the sort of 
cosmic abundance of Helium. 
Where did the 25% of the universe that is 
Helium come from? 
And this becomes clear in a project 
undertaken by George Gamow and his 
student, Alpher, in 1948 and they 
actually, Gamow's agenda, the time 
stellar dynamics was not nearly as well 
understood as it is today, Gamow's agenda 
was to show that back in the hot, dense, 
early universe all of the elements were, 
in fact synthesized and that the current 
cosmic abundances could be explained by 
the Big Bang. So, what's the idea? If you 
go far enough in the past, there was a 
brief episode in the early history of the 
universe where temperatures were high and 
densities as high as in the interior of 
stars. 
So, there was fusion going on everywhere 
at every position, 
right here, what is the position that is 
right here, right now, a long time ago 
was a part of a very hot, dense plasma 
where fusion was going on. 
So, fusion was going on pervasively, and 
Gamow's idea was that this is where 
heavier elements than hydrogen were 
created. 
It turns out that, remember that the 
crucial step for synthesizing heavy 
elements was the triple alpha process. 
the triple alpha process worked in the 
context of a star that had long enough 
time to maintain a stellar core as high 
density for a while. 
only very rapid processes work in the 
context of Big Bang nucleosynthesis 
because very rapidly, the world cools and 
dilutes, and densities decrease, you 
don't have the stellar envelope holding 
things in, but it does correctly predict 
the Helium abundance, let's see how that 
works. 
So, here's the assumptions. We're going 
to assume, that reactions maintain 
thermal status of thermal equilibrium in 
the gas, this hot dense plasma, 
in the context of an expanding flat 
universe. 
This means that the reactions proceed 
rapidly enough that before the universe 
has diluted things away 
chemical abundances can adjust. And, of 
course, at some point, it stops and that 
is the, at that point, you sort of freeze 
in the situation that you have. 
So, let's see how that works. 
remember that at high temperature, we 
have relativistic particles and the 
number density of a relativistic particle 
is simply given by the Stefan-Boltzmann 
Law. 
The energy density of a relativistic gas 
is proportional to t^4, the energy, 
average energy of the particles 
proportional to t, so the number density 
of relativistic particles is proportional 
to kT^3, and in particular, is 
essentially independent of their masses. 
On the other hand, at the low energy, at 
low temperature we have non-relativistic 
particles. 
And if you imagine that there are 
processes that can create all kinds of 
particles 
then you can think of particles as energy 
states, as and just in the way that in a 
gas with temperature T, there can be a 
tail, remember, of particles, Helium 
particles, whose speeds are 10 times the 
average thermal speed but that's an 
exponential decaying tail. 
the similar process holds in creating 
particles of the, thermal equilibrium 
where there are processes that can create 
and destroy particles at temperatures 
below where kBT is less than mc^2 the 
number density of a particle decreases 
exponentially with the ratio of rest 
energy to the thermal energy, particles 
that are way massive than the pervase, 
more massive than the pervasive 
temperature are unlikely to be found and 
as this exponent approaches 0, you'll 
exactly go over to the relativistic mass 
indpendent density. 
So, whose radiation? Well, that depends 
on your temperature. I remind you of our 
list of particles. 
I've slightly increased it from back when 
you were talking about the sun, you see 
their charges. 
And based on their masses, I can tell you 
that protons, for example, become 
relativistic of temperatures of about 
10^13 K. 
And this g is that geometric factor, the 
number of degenerate states that a 
particle has that relates how many of 
these there are to how many photons there 
are. 
So, that's an extra factor of our 
[UNKNOWN] density. 
protons become relativistic at 10^13 
Kelvin. 
What about neutrons? 
Well, neutrons are a little bit more 
massive than protons so they become 
relativistic at a temperature that is 
just a little bit higher 10^13, but the 
main important difference is that the 
temperature difference for neutrons to 
become relativistic is about 1.5*10^10 K. 
Electrons, being much lighter become 
relativistic at about 6 billion Kelvin. 
neutrinos, as far as we know, are 
massless. 
They can become relativistic at any 
temperature. 
Certainly a photon, which is exactly 
massless is relativistic at all, 
temperatures isn't always moves at the 
speed of light. 
muons, intermediate mass become 
relativistic at about a trillion K. 
And another kind of particle called the 
pion, this is not the right number and 
the mass of a pion is 140 some, 2 maybe 
MeV and so they become relativistic at 
about 1.6 trillion degrees, there's 3 
different kinds of them with three 
different charges and this is a relic 
from an old table. 
okay, so there's a collection of 
particles and as temperatures increase 
moving into the past, more and more of 
them are relativistic. 
So, what has this got to tell us? 
conversely as temperatures cool, less and 
less of them are relativistic. 
So, let's start at the beginning with a 
very hot universe with a temperature 
above a trillion Kelvin. 
neutrons and protons are marginally 
non-relativistic but of all the species 
that are relativistic, then you have 
thermal equilibrium. And in thermal 
equilibrium, as we'll see, 
the processes of Physics do not 
distinguish particles from antiparticles 
so there is as many antielectrons as 
there are electrons. 
There is as many antiprotons as protons, 
there are almost as many. 
So, there are large quantites of 
antiprotons and protons, antineutrinos 
and neutrinos, antimuons and muons, and 
so on. 
the ratio between the number density of 
neutrons and protons is given by the 
ratio of those two exponentials which is 
proportional to the difference of their 
masses in units of kT and since the 
difference in their masses is small 
compared to the temperature even though 
they're not relativistic, there's about 
the same number of neutrons as protons. 
Now, as the universe cools by about a 
factor of 10, 
by the way, these temperatures obtained 
for the first ten thousandths of a second 
after the Planck time, so we're deep into 
the radiation dominated era and all of 
the exciting events occur very early in 
the history of the univesre. 
by that time temperatures have decreased 
by about 10^11 Kelvin. 
muons stop being relativistic. 
When muons slow down, these muons and 
antimuons find each other, they 
annihilate. 
most of the muons are gone, only a small 
remnant of actual muons survive. 
The excess muons over antimuons, we'll 
talk about that later, and the muons are 
gone by this time. 
If you plug into this expression there's 
still equilibrium, the number of 
neutrons, ratio of neutron number to 
proton number is about 0.86 and note 
[COUGH] that neutrons, unlike protons, 
are unstable. 
Of course, 
they decay. 
But they decay in 15 minutes. 
Neutron decay is not playing a 
significant role yet. 
It will later. 
as the temperature cools further by the 
time we get to 30 billion K at time about 
a tenth of a second after the Big Bang 
neutrinos, because of the slow rates of 
the weak interactions essentially 
decouple at this point neutrons or 
neutron to proton ratio has decreased to 
about 0.6. And by the time you get to 5 
billion Kelvin at a few tens of seconds 
electrons have become, are beginning to 
become non-relativistic, electrons and 
positrons annihilate, producing a lot of 
photons. 
Notice that the, the neutrinos are, are 
already decoupled. 
So, at the time that the neutrinos 
decoupled, of course, neutrinos and 
photons were all strongly interacting. 
There were two gases that could exchange 
energy, they had the same temperature. 
So, do we expect a gas of primordial 
neutrinos, just as there are cosmic 
microwave backgrounds, sort of, except 
that when all these electrons annihilated 
electron and positron produce two photons 
with energy 511 KeV each. 
And so this produces more photons heating 
the photon gas injecting energy into the 
photon gas that is then rethermalized. 
And for that reason, the photons at that 
point are harder than neutrinos. 
From that moment on after recombination, 
photons evolve the same way as neutrinos, 
they are temporarily, they remain 
blackbody and their temperature decreases 
with the scale factor. 
So, the ratio between photon temperatures 
and neutrino temperatures was in, sort of 
implanted at the time of electron 
annihilation. 
And the neutrinos are colder than the 
photons by the factor of 1.4, so there's 
less energy in neutrinos than there is in 
photons, but there is definitely an 
ambient cosmic neutrino radiation flying 
around. 
But that's not what we're about. 
We're talking about the neutrinos, about 
the neutrons and the protons because 
that's what it takes to make Helium. 
Notice that unlike in the present 
universe, in the present universe, you 
mostly find protons, neutrons only exist 
in nuclei. 
That's because they have billions of 
years to decay. 
Free neutrons do not exist. 
At these short times, neutrons have not 
decayed yet. 
Now, back when the universe was hot and 
dense the protons and neutrons are in 
chemical equilibrium, 
there are these rapid reactions that can 
convert neutrons to protons and protons 
to neutrons, in the presence of this sea 
of electrons and neutrinos, as the 
temperatures cool and the neutrinos 
decouple. 
And soon thereafter, most of the 
electrons and the positrons disappear. 
this reaction slow down by that time the 
neutron to proton ratio is decreased to 
about 0.22 because at this temperature, 
the mass difference is significant, it's 
either minus 1.5. Now, notice that once 
this ratio is achieved and temperatures 
have decreased, 
this ratio is frozen in essentially 
because reactions converting protons to 
neutrons and back, and vice-versa, no 
longer take place. 
So, from this moment on, these are the 
neutrons that you have. 
Now so now, you have a gas of neutrons 
and protons and, of course, over time, 
the neutrons decay so you have less and 
less neutrons. 
However, the universe is cooling and 
expanding and cooling, and when 
temperatures cool down to about a billion 
Kelvin, that is a time of about 180 
seconds after the Big Bang, Deuterium is 
stable. 
So, if a proton and a neutron happen to 
be moving slowly enough next to each 
other and bind to form Deuterium, 
Deuterium will not be blown apart, 
photodisintegrated, by high energy 
photons once the energy of the photons is 
less than a billion kelvin. 
So until that time there was no Deuterium 
because if Deuterium, formed it would of 
been blown apart by a photon after 180 
seconds after the Big Bang, Deuterium can 
form. 
Now, at this point essentially all of the 
neutrons that are left form Deuterium. 
How many neutrons are left? Well, they 
had a fraction of about 0.223 of neutrons 
to protons, but 180 seconds have gone by. 
The half-life of a neutron these are free 
neutrons, is about 600 seconds. 
you plug that into our half-life 
radioactive decay formula, remembering 
that neutrons decay to protons so it's 
parent to daughter ratio the number of 
neutrons has decreased, protons 
increased, The ratio has approximately 
halved, by this time, only about 12% of 
the baryons in the universe are neutrons. 
And now, once Deuterium is stable a 
neutron inside the Deuteron is stable, 
neutrons stop decaying. 
Essentially, all of the remaining 
neutrons are bound up in Deuterium and 
very rapidly thereafter, they formed a 
very stable alpha particle nucleus. So, 
essentially, all of the neutrons that 
survive produce Helium. 
And so, we get left with a Helium nuclei 
and three protons and no more neutrons. 
So, what does this predict about how many 
Helium nuclei you will have formed? Well, 
you can figure it out. 
how many of the nuclei that we have left 
are Helium nuclei? Well, each Helium 
nucleus requires 2 protons, 2, and 2 
neutrons, since we had this many 
neutrons, this was our neutron to proton 
ratio, divide that by 2, that's sort of a 
count of how many neutrons how many 
Helium nuclei were created. 
How many particles 
what is the total number of particles? 
Well, we had a sort of the total number 
of neutrons plus protons. 
Add that up to 1 but then each new, each 
formation of a Helium nucleus took away 2 
protons and took away 2 neutrons. So this 
many this many Helium nuclei took away 
four particles, replaced them by one so 
each Helium nucleus reduced the number by 
three. 
Plug in these numbers, you find that 7.5% 
of all the particles that are left in 
your all the baryons that are left are 
Helium nuclei and, of course, the others 
are protons. 
And any free neutrons that were left 
around that did not get stopped in, will 
rapidly within, you know, a few minutes, 
decay and so you'll, 
you'll end up with protons and alpha 
particles, that's all you'll end up with. 
So this is the fraction of the universe 
that you predict by this very rough 
calculation, will be Helium and in terms 
of mass, since the mass of a Helium 
nucleus is 4 times the mass of a proton, 
the mass fraction of Helium that you 
predict is about 30%. 
this is not bad. 
Remember, what we observe is about a 
quarter of the mass of the universe is in 
Helium. 
This very rough calculation produces an 
result that is very close to correct, 
a more refined calculation produces 
agreement. 
Notice that since essentially, all I 
needed to know was the ratio of neutrons 
to protons, it's very insensitive to 
details. 
the rate at which the universe cools, 
it's a radiation dominated universe, that 
has to do with the density of photons in 
the universe which I know because I 
measure the cosmic microwave background 
radiation. 
I know the energy density in the thermal 
microwave background. 
Today, I can extrapolate that by energy 
conservation back to those times. 
There is very little ambiguity. 
There's no free paramiters. 
And then, that get, that sets the time 
rate for all of these reactions to occur 
and basically nothing depends on any of 
the details the fraction of the universe, 
the protons and the neutrons that are 
converted to Helium atoms is a very 
robust prediction. 
Okay. So, we can understand where Helium 
came from. 
But about other nuclei, remember, gamma 
was trying to produce all nucleons, all 
nuclear, nuclear species. 
It turns out that fusion to heavier 
nuclei than Helium is very inefficient, 
but you form some small fraction of trace 
amounts of lithium and excitingly a 
little bit of Helium 3 and excitingly a 
fraction of, a small fraction of 
Deuterium. 
So, some of the deuterons did not fuse to 
form Helium and how many of them did not 
fuse to form Helium depends on the speed 
in which the universe expands driven by, 
remember, the density of the photon gas, 
relative to the density of nucleons 
because the denser nucleons are, the more 
rapid nuclear reactions are. 
So the abundance of Deuterium is very 
sensitive to this ratio of the density of 
baryons to the density of radiation and 
since today, density predicts the density 
then you can put constraints on the 
baryon density relative to the density of 
radiation, which remember, is known 
because this is basically sigma times 
2.7k^4/2c. 
I know this. 
Well, this is rho radiation at current 
time. 
The background radiation gives me omegaR, 
precisely. 
So, this gives me great, strong 
constraints on the baryon density. 
This, in the sense, tells us two things. 
one is it tells us that not more than 5% 
of the total energy of the universe or 
the critical density could possibly be 
baryons, else we would have been left 
with lots more Deuterium. 
And I should point out that there are 
very few processes after the Big Bang 
that are likely to form Deuterium without 
producing alpha this is a a process of 
Nuclear Physics. 
and furthermore, it tells us that so, so 
that if we find that the density of dust 
of matter, non-relativistic matter in the 
universe, is larger than that 5%, then 
the remaining amount must be 
non-baryonic, and therefore, this is the 
25% of the universe, that we think is 
composed of dark matter. 
This is one of the strongest constraints. 
you can, of course, do some more refined 
calculations that take into account both 
the concentration of Helium and of 
various isotopes and as I said, the trace 
amount of Lithium that was produced and 
so, this gives us a constraint on 
Cosmology. 
Moreover, if you tweak your model of 
particle Physics by all kinds of 
additions to the standard model trying to 
understand as we will see theories beyond 
the standard model, 
they might make small modifications to 
the rates of these nuclear reactions. 
things are extremely sensitive to the 
rates of nuclear reaction. 
For example, you can do a rather straight 
forward calculation that shows that if 
there were more than three, remember, 
there were three species of light 
neutrinos the muon neutrino, the electron 
neutrino, and the tau neutrino. 
more than three species of light neutrino 
would lead to unacceptable to Deuterium 
concentrations that are incompatible with 
the observation and so, in that sense, we 
know there cannot be a fourth as yet 
undetected neutrino species because it 
would ruin Big Bang nucleosynthesis, a 
very sensitive test of our understanding 
both of Cosmology and of particle Physics 
and one of the great successes of Big 
Bang Cosmology