### The 'Inflation Debate' in Cosmology Continues

Last year Scientific American published a controversial article titled "Pop goes the Universe", the three authors Anna Ljjas, Paul Steinhardt and Abraham Leob are skeptics of inflationary cosmology and they gave their reasons why. That was followed with a letter signed by 33 physicists who study cosmology, denouncing the article that had been published.

Originally I didn't even want to comment, the whole business of signing a letter denouncing the article instead of writing to the editor rubs me the wrong way. It reminds me when a hundred Nazi scientists were called on to denounce Einstein, if he were wrong you'd only have needed one. Science media should be an open space where people are allowed to hear about controversies in physics. Inflation is not a well established scientific theory like the Big Bang or evolution, and some very highly respected theoretical physicists are in real doubt.

Inflation probably is the correct theory, it's the best worked out model to date but I still have some worries about it. Some of the arguments used in favour of inflation aren't convincing, at least to me. For one, inflation is supposed to explain the observed large scale homogeneity and isotropy in the universe and it does. But virtually everyone agrees it re-introduces the same problem by assuming an even more homogeneous finely tuned initial patch at the start of the universe, $P= e^{S}$ where $S =1/\Lambda$ in fact its orders of magnitude worse.

Second the known relation between the mass density of the universe and it's critical density, $\Omega _{tot} \equiv \frac{\rho _{tot}}{\rho _{c}}$ we know that the curvature density parameter increases with time, so that it must be smaller in the past. The exact numbers don't matter but this essentially is the "flatness problem" because cosmologists expect the initial value to start out approximately at one and inflation is supposed to reduce it to something compatible with observation.

I don't know of any reason why the universe should start off with a value of curvature density close to one, if it started out with something much higher inflation would give you the wrong answer. But even if we insist it did start out at one, inflation forces you to pay a price, it postulates the existence of an undetected new field, a potential for that field and a new particle. None of which we can observe because the field has since decayed into our Hot Big Bang universe.

Thirdly inflation predicts a statistical distribution of anistropy in the universe, which we can measure and inflation can be modeled to fit the data well, although you could adjust the energy density curve in the theory to fit virtually any observation. It does have alternatives though, the Ekpyrotic Cyclic scenario can also explain the data, String Gas Cosmology (though not well developed and may have a "flatness problem" too), Penrose's Conformal Cyclic Cosmology and Cosmic Strings are also commonly proposed as alternative ideas.

On this same point, inflation is in high friction with data from the Cosmic Microwave Background Radiation, the Planck satellite which measured the CMB, a relic from when matter decoupled from radiation, found temperature anisotropy variations are scale invariant to 0.01 of a per cent. This is incompatible with the simplest models of inflation; the standard text book model you will have learned if you took any classes in physics.

Sometimes it's misleadingly described as "evidence for a multiverse" or eternal inflation because proponents of inflation are forced to retreat and consider more complex mechanism for the decay of the inflaton field, like those involving a slow roll potential or plateau. Though convex potentials are still allowed by the Planck data.
Not only do these more complicated plateau models need extraordinary fine tuning, both in the potential they artificially construct but also in the initial entropy of the Big Bang which the theory, by itself cannot explain. Moreover, neither in the Planck result nor any experiment since has there been any detection of primordial gravitational waves, if they had I wouldn't be writing this post and I would change my position. These waves are ripples in the fabric of spacetime which are caused by the exponential stretching of space during the inflationary epoch, and we'd expect to find them if the theory is correct.

The thing that worries me the most about inflationary cosmology, is by extrapolating known physics to vastly larger orders of magnitude, it's now promoted the multiverse to forefront research in theoretical physics. I don't have any a priori problem with the existence of multiple universes but the idea of modern science decoupling from rigorous scrutiny and an emphasis on empirical evidence worries me. I don't know if there are other universes - I wish I did - but unless a proposal is testable and falsifiable, it's not science.

To be sure, there are some versions of the multiverse which may lead to observational consequences for our sector of it, things like bubble collisions with other bubble universes or entanglement between the early universe and others leaving some mark on the inflaton's potential in our universe. But so far no evidence has been found, despite the fact there have been genuine attempts to look for these signatures. Though if you're a proponent, you hardly need to worry about "what isn't there", so long as you have an infinite multiverse, anything that happens you can explain away as occurring in some sector.

There are a whole load of other reasons to be skeptical of eternal inflation, like the measure problem that I've talked about when I reviewed Lawrence Krauss' book "A Universe From Nothing".

### 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…