### 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.

### Set Theory

This post is a very brief introduction to some of the basic concepts of set theory. Set theory is a branch of mathematical-logic, that has wide applications across disciplines. Its not just used in the obvious way of studying the foundations of mathematics by mathematicians but also in physics, social science, and even by philosophers as a theory of semantics for predicate logic (although you can do propositional logic without set theory).

A set is a collection of elements, or members; the notation for a set is specified by listing its components. So the set of even numbers can be represented a
$E: \left \{ 2,4,6,8 ... \right \}$$E: \left \{ x: x > 0 \wedge even\right \}$ Either of these notations is valid. Further, elements of a set can only be in that set, once. So   $E: \left \{ 2,2,2,4,4,6,8 ... \right \} = E: \left \{ 2,4,6,8 ... \right \}$ The notation used to indicate that something is an element of a set, is using the Greek symbol "epsilon". That is: $4 \epsilon S$…

### 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&#…

### Can inflation be eternal into the past?

Back in 2003 a paper appeared on the arXiv titled "Inflationary spacetimes are not past complete" that was published by Arvind Borde, Alan Guth and Alexander Vilenkin which has had considerable amounts of attention online. The theorem is rather uninteresting but simple and doesn't require a very complicated understanding of math. So I thought I'd explain the result here.

It's purpose is to demonstrate that inflationary models are geodesically incomplete into the past which they take as "synonymous to a beginning" but Vilenkin stresses that the theorem can be extended to non inflationary models so long as the condition of the theorem that the average rate of expansion is never below zero is met. These models too then are incomplete into the past. Consider the metric for an FRW universe with an exponential expansion

Where the scale factor is

Since the eternal inflation model is a "steady state cosmology" the mass density and the Hubble paramet…