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