Normalization of Power Spectrum
Recall we had the power spectrum:
is set in the primordial universe, and we have no way of theoretically predicting , the normalization constant. We have two choices for getting :
- Use CMB data
- Use galaxy clustering strength
The variance of linear perturbation on the scale of (or ) is given by the equation:
where is some window function, and is the Fourier transform of that window. In general, people have decided to normalize this to an average over the scale of , and to denote this as .
Recall from last time that is the inflaton field. The energy density and pressure from are given by:
For this pair of equations, if and , then . We found that this means , and we have , so we have an exponentially expanding universe.
Our equation of motion reads:
So what is ? Typically , where is the temperature. At finite temperature, The effective potential is related to the potential by:
For , , and when . This non-zero potential at is called the false vacuum.
If , we have the same value of at , but the potential decreases to a minimum at , where the true vacuum is. After that point, ; it increases again. Thus, the relationship between and is governed by the relationship between and . Remember that in order to have inflation in the early universe, we need . Therefore, as we have a gradual roll-off from the false vacuum into the true vacuum, we have inflation.
The End of Inflation
Inflation stops when the slow roll-off doesn’t hold anymore (). As the universe makes the phase transition from the false vacuum to the true vacuum, will oscillate about the minimum vacuum energy at . In our Lagrangian, is coupled to radiation, so this oscillation leads to a “reheating” of the radiation in the universe. This is effectively a frictional term on , which allows to finally settle into the true vacuum. Adding this friction term to our equation for the evolution of (and setting –we’ll assume the universe is spatially homogeneous) we have:
Similarly the evolution of the radiation density is modified by this interaction:
Origin of Fluctuations
Quantum fluctuations in the vacuum field can cause different portions of the universe to undergo the phase transition into the true vacuum state at slightly different times (). This causes different portions of the universe to undergo heating at slightly redshifts. Areas which undergo heating at earlier redshifts will thus end up a little cooler at the time that inflation ends. We can show that , the potential fluctuations, is given by:
where is the density fluctuation. Thus, , so:
From messy QFT calculations, we can say :
So constant. This means , and so . However, it should be noted that although is constant, is not perfectly constant, and so doesn’t scale perfectly as .