### Chaotic Inflation Violates Penrose-Hawking Singularity Theorems

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

If the pressure of matter p is positive then the rate of expansion is either decreasing or the rate of contraction is increasing but if the pressure is negative and the early universe is pervade by tension (a negative gravitational force) rather than normal pressure, then the expansion rate would be accelerating.

Strictly speaking inflation refers to when the scale factor was accelerating.

From this we can see immediately that inflation violates the strong energy condition. Since it requires the inequality, to hold.

This gives us only a basic understanding of cosmic inflation but there are at least twenty-five or so variations of the theory, old inflation, news inflation, eternal inflation, and so on. The kinds of models cosmologists use today, to avoid the initial singularity are fairly sophisticated.

This here is a model of chaotic inflation, proposed by Andrei Linde in 1983, where

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.

$N(t) \equiv ln[a(t_{end})/a(t_{beg})]$

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 initial patch of energy required to start inflation is incredibly small and increases exponentially with the volume of the universe.

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.

### William Lane Craig and the Hartle-Hawking No Boundary Proposal

Classical standard hot Big Bang cosmology represents the universe as beginning from a singular dense point, with no prior description or explanation of classical spacetime. Quantum cosmology is different in that it replaces the initial singularity with a description in accord with some law the "quantum mechanical wave function of the universe", different approaches to quantum cosmology differ in their appeal either to describe the origin of the material content of the universe e.g., Tyron 1973, Linde 1983a, Krauss 2012 or the origin of spacetime itself e.g., Vilenkin 1982, Linde 1983b, Hartle-Hawking 1983, Vilenkin 1984.

These last few proposals by Vilenkin, Hartle-Hawking and others are solutions to the Wheeler-DeWitt equation and exist in a category of proposals called "quantum gravity cosmologies" which make cosmic applications of an approach to quantum gravity called "closed dynamic triangulation" or CDT (also known as Euclidean quantum gravity). I&#…

### How Should Thatcherites Remember the '80s?

Every now and again, when I talk to people about the '80s I'm told that it was a time of unhinged selfishness, that somehow or other we learned the price of everything but the value of nothing. I can just remember that infamous line from Billy Elliot; 'Merry Christmas Maggie Thatcher. We all celebrate today because its one day closer to your death'. If it reflected the general mood of the time, one might wonder how it is she won, not one but three elections.

In an era when a woman couldn't be Prime Minister and a working-class radical would never lead the Conservative party, Thatcher was both and her launch into power was almost accidental owing in part to Manchester liberals and the Winter of Discontent. Yet I'm convinced her election victory in '79 was the only one that ever truly mattered. Simply consider the calamity of what preceded it, the 1970s was a decade of double-digit inflation, power cuts, mass strikes, price and income controls, and the three…

### Creation Of Universes from Nothing

The above paper "Creation of Universes from Nothing" was published in 1982, which was subsequently followed up in 1984 by a paper titled "Quantum Creation of Universes". I decided it would be a good idea to talk about these proposals, since last time I talked about the Hartle-Hawking model which was, as it turns out, inspired by the above work.
Alexander Vilenkin also explains in a non-technical way the essential idea in his book; Many World's in One – one of the best books I've ever read – it mostly covers cosmic inflationary theory but the 17th chapter covers how inflation may have begun. In fact Vilenkin is one of the main preponderant who helped develop inflation along with Steinhardt, Guth, Hawking, Starobinsky, Linde and others.
Although I won't talk about it here, Vilenkin also discovered a way of doing cosmology by using something called "topological defects" and he has been known for work he's done on cosmic strings, too.
In ex…