Claim: Chaotic Inflation can avoid the implications of the Penrose-Hawking singularity theorem
In the last post I talked about the Penrose-Hawking singularity theorem, and the conditions under which the theorem can be violated. One of the assumptions that Penrose and Hawking made was the strong energy condition, which states
This is true when the cosmological constant of the universe is zero, this has been a major weakness in their argument because neither the theorem nor the Friedman equation can tell us what the material or field content of the early universe was which has a direct affect on the pressure, mass density and expansion rate of the early universe. Under the second Friedman equation
Strictly speaking inflation refers to when the scale factor was accelerating.
We assume the universe started off with a sufficiently large scalar field, so that the Hubble constant is proportional to the energy density of the universe. We also have a simple equation for the value of the scalar field, itself.
This is the Klein-Gordon equation and it behaves very similarly to the equation for simple harmonic motion. If you have a large scalar field, then H will be large and the scalar field potential reduces very slowly. Then the Hubble constant is nearly constant.
Finally, that would imply
This tells me that the scale factor is accelerating. This likely happens around the GUT era of the Big Bang and after about ~ 100 of these e-folds the universe is approximately the size of a marble and starts to decay. An e-fold is a logarithmic measure of how large the universe grows during inflation.
One might wonder what happens to density in the Friedman equation during the inflationary era, as so happens density is conserved, so that pressure must be negative. Large amounts of positive and negative energy appear spontaneously out of the vacuum. So that the total amount of energy is a function of time
The argument that chaotic inflation can avoid the Penrose-Hawking singularity is very convincing, but this doesn't necessarily mean it avoids a singularity altogether. In an up coming post I'll discuss whether eternal inflation can really avoid a beginning of time.