Good evening everybody. I have to look in the camera now because yesterday even though

we had the homework we did do 45 minutes of lecturing so if somebody wasn't there yesterday

they missed one hour of lecturing. In brief what we did, so don't be confused, in brief what we did

was last week we had looked at thermodynamics and had drawn the conclusion that in order to calculate

what parts of the universe are relevant for constituents of the universe are relevant for

its evolution that only those matter which really are in thermodynamic equilibrium defined as those

particles for which the reaction rate for interactions with the particles was faster

than characteristic time scales of the universe which are characterized by the current Hubble

parameter and we have then taken a quick look at what at the way how relativistic thermodynamics

works it started out with the standard thermodynamical equation for the distribution

momentum distribution of particles and we had then inserted this into the standard equations

right knowing the distribution we can calculate particle number density the energy density and

then also the pressure and then we had inserted the energy equation which we were allowed to do

because we have a very symmetric distribution in momentum and if you write momentum as a function

of energy as given by this equation down here you see that there are times either where the energy

is essentially dominated by MC squared or where it's much larger and depending on which of these

cases we have we either have very relativistic particles or non relativistic particles and doing

these integrals is a bit dry as some people complained afterwards and they're right the

important outcome of these calculations and we forgot the books that we wanted to bring

we yesterday had a slight aside about what integral tables look like and that indeed you

can look up integrals in books rather than online and I had promised to bring books which we did not

bring but the conclusion of all of this was that depending on whether you have bosons or fermions

in principle these particles all behave the same it's just that you have a rather annoying factor

of three-quarters or seven-eighths in the calculation of particle density as a function

of temperature or energy density as a function of temperature which makes life more difficult

but in the end what we learned was that if you want to calculate the overall energy density you

just sum over all particles weigh their statistical weights in a proper way as given down here by

equation 1054 and then you effectively have the energy density of the particles in addition and

finally we calculated what the early expansion of the universe looked like namely what we did

was we calculated how the temperature in the early universe depends on time and we could do

that because we had seen that particles have a relativistic distribution which means that their

density must scale with a scale parameter to the fourth power because all relativistic particles

behave the same irrespective of whether they are photons with something else and the outcome was

that we managed to on the one hand calculate the time dependence of the Hubble parameter and on the

other hand we calculated the temperature dependence of the Hubble parameter as a function of time and

by combining these two things we got a relationship between the temperature and the time which is down

here and the only unknown in that relationship is this parameter g star which is the weighted sum

over the degrees of freedom of the individual relativistic particles okay and so in order to

convert that into something that we can use well we just have to look at what particles are there

and when are they at what times what temperatures are they relativistic well they are relativistic

whenever kT is much larger than mc squared so if you just make a table of the particles write down

their energies here or the energy equivalent as and and then you can immediately just sort this as a

function of temperature and by summing up the g's and you can calculate g star and that in the end

tells you how as a function of temperature g star changes the one uncertainty here is the confinement

temperature of quarks because there's a point when quarks are either free or not and depending on

this on that energy g star changes and that's given by this blue line so effectively what we

have is that the early universe which was very hot was a soup of many many many relativistic

particles and then as the universe expanded the universe cooled down and whenever the

temperature went below the threshold energy for particle to switch from relativistic to

non relativistic right it's not really a switch there is a time in between obviously but whenever