BGV and KCM

OK, so last time I claimed that our best, experimentally successful model of the early universe is one that is infinitely old and has no initial singularity. If you are savvy about these things (from listening to Craig debates, for instance) you are wondering, "But what about the Borde-Guth-Vilenkin theorem? Doesn't it prove that an inflating universe must have had an initial singularity?" The answer is, "No, it doesn't."

There has been a lot of confusion about this, with clip quotes from one or another of the paper's authors being traded to "prove" that the theorem does, or doesn't, prove the universe had a beginning. So let's look at what the theorem actually says.

"Any theorem is only as good as its assumptions," writes Alexander Vilenkin in a letter to Lawrence Krauss. So let's start with the assumptions of the theorem. Roughly speaking, there are two:

1. Spacetime is classical.
2. Spacetime is expanding on average.

(I say "roughly speaking" because there are all sorts of technical issues about spacetime congruences and so forth that one needs to make these assumptions precise, but I think these short versions are sufficient to understand the main philosophical issues involved.)

Now let's jump to the conclusion, which I quote from their paper:

... we see that if Hav > 0 along any null or noncomoving
timelike geodesic, then the geodesic is necessarily
past-incomplete.

Some translation: a "timelike geodesic" is simply the path that an object will travel on if it is not subject to any forces other than gravity. Similarly, a "null geodesic" is the path that a light ray will travel. "Hav > 0" is the mathematical statement of assumption (2.): the universe is expanding on average. "Past-incomplete" means that if you try to follow one of these paths backwards in time, you can only do so for a finite amount of time.*

OK, so here's my first point: the BGV theorem is not a singularity theorem!

The conclusion says nothing at all about singularities: it only says that certain paths cannot be extended infinitely backward in time. One way that this might happen is if the path encounters a singularity. But that is not the only way it can happen.

The other thing that can happen is that, as we trace the path backward in time, we encounter a region where one (or both) of the assumptions of the theorem is no longer valid.

Start with assumption (2.). The path can enter a region in which spacetime is static, or contracting, or cyclically expanding and contracting. Then assumption (2.) is violated and the theorem's conclusion is avoided. In spacetimes like these, the BGV theorem simply doesn't apply.

What about assumption (1.)? Classical spacetime is a pretty basic assumption in any sort of cosmology. But it is expected to break down in the quantum gravity regime, where we encounter "spacetime foam" of some sort. This is difficult to discuss, since in the absence of a good theory of quantum gravity, no one has any idea what spacetime foam should look like. But it's possible that the BGV theorem is pointing us to a place where quantum gravity comes into play.

So the BGV theorem does have something very interesting to tell us about the early universe: namely, that the infinitely old, infinitely expanding inflationary period that I discussed in the previous post cannot be the end of the story. Not just because of vague speculations about the Planck epoch, but because of the properties of classical spacetime itself. Here's what the authors said in the paper itself:

Whatever the possibilities for the boundary [where the geodesics come to an end - RNO], it is clear that unless the averaged expansion condition can somehow be avoided for all past-directed geodesics, inflation alone is not sufficient to provide a complete description of the Universe, and some new physics is necessary in order to determine the correct conditions at the boundary. This is the chief result of our paper.

But the BGV theorem does not say that spacetime must be singular; still less does it say that there was an initial singularity from which the entire universe arose. This is why I say that the theorem is irrelevant to the Kalam Cosmological Argument: it doesn't say anything about whether the universe had a beginning or not.

* A few more technical points: by "backward in time" I mean in the opposite time direction from the direction in which the universe is expanding. And by "finite amount of time" I mean time as measured by a clock carried along with the moving object (proper time). In the case of null (light) rays, we have to use an "affine parameter" rather than the proper time.

The Universe Is Infinitely Old (Says Cosmology)

The Secular Student Alliance at my school recently  hosted a debate on the topic "Is there a God?" The theist side was very well prepared and did a great job in the debate, as even the secular students in the audience agreed. One of the arguments they presented was the Kalam Cosmological Argument, which was presented rather the same way that William Lane Craig presents it. The discussion brought up the Borde-Guth-Vilenkin Theorem (BVG for short), which Craig has used as well, as support for the premise "The universe began to exist." I want to talk about why the BVG theorem is irrelevant to the Kalam Cosmological Argument, but first, by way of preliminary, I want to discuss the current state of cosmology.

Cosmology begins with Einstein's equations of General Relativity (GR for short), and asks whether these equations, applied to the universe as a whole, are capable of explaining what we see when we look out into deep space. GR relates the curvature of spacetime to the energy content in the universe, so in order to solve the equations we need to know what the universe is filled with. There are three basic types of energy we need to consider:

• Matter in the form of galaxies, dust, dark matter, and the like,
• Radiation, including particles moving so fast that they are relativistic, and
• Cosmological constant, aka "dark energy."

The three forms of energy behave differently, so different ones are important at different times in the evolution of the universe. For most of the last 13.7 billion years, the expansion has been matter-dominated. But in the far future, the expansion will be dominated by the cosmological constant, and at very early times (the first 50,000 years or so), the expansion was dominated by radiation.



This plot shows the "scale factor" - roughly speaking, the size of some patch of the universe, as a function of time.The universe described by this model fits extremely well with the observations of distant galaxies, supernovas, quasars, etc.

If the early universe is indeed radiation dominated, then the scale factor goes to zero at some finite time in the past: that is, there is a Big Bang - an initial singularity.

However, we now have an alternative account of the earliest moments of the universe. Inflationary cosmology, proposed by Alan Guth in 1980, then in a corrected from by Linde and (independently) by Albrecht and Steinhardt in 1982 supposes that before the radiation-dominated epoch there was another epoch, dominated by a cosmological constant - but a very much larger cosmological constant than the one we measure now. I'm not going to go into the reasons these physicists thought there might have been a very large cosmological constant in the early universe, which then "switched off" (meaning it wasn't really a "constant", obviously): you can read about it at the Wikipedia page if you're interested.

The inflationary model was able to explain several features of the universe that had been puzzling in earlier cosmological models: the flatness problem, the horizon problem, and the monopole problem. In science, though, explanatory power is not enough for a theory to become accepted. In addition, a theory has to make novel predictions that are confirmed by experiment before scientists accept it as (likely to be) true.

(I can't help pointing out how different this is from theistic "explanations," in which God is claimed to be the explanation of things like life, morality, or the universe, but where there is no concern for making testable predictions about these realms.)

We now have several good reasons to think that there was in fact such an inflationary epoch. One of these is the pattern of fluctuations of the cosmic microwave background, that fits extremely well with the predictions of inflation:



Another is the very recent BICEP2 result, that seems to show the effects of quantum gravity on the polarization of the cosmic microwave background, in a way consistent with the predictions of the inflationary model.

So inflationary cosmology replaces the initial singularity with a period of exponential expansion.



How long is this inflationary period? Well, an exponential function never reaches zero, so the inflationary period is, in principle, infinitely long!

Let's sum that up: According to our best, experimentally verified model of cosmology, the universe is infinitely old and has no initial singularity! 

This is the current state of our understanding of the early universe. Now, I have to admit right away that no physicist thinks the inflationary model is the end of the story. The exponential expansion is so fast that in a very short time the scale factor reaches the Planck realm, where we expect GR to break down and quantum gravity to come into play. So most diagrams of the early universe insert a quantum gravity region before the inflationary epoch. (In this diagram from Andrei Linde it's labeled "Space Time Foam.")



There has been much discussion of what went on before the inflationary epoch: quantum foam, the no-boundary proposal, the cyclic universe, and so on. There is even a version called "eternal inflation," in which some portion of the universe goes on inflating forever, while pocket universes like ours bubble off from time to time. In some of these models, time is finite in the past. In others, it is infinite. But all of them are pure speculation: there is to date no experimental confirmation of any of these scenarios.

Does modern cosmology support the premise that the universe had a beginning? Emphatically, no! Our best model extends infinitely into the past, with no initial singularity. We know better than to take that prediction as the last word: likewise, we know better than to take models that do exhibit an initial singularity as the last word. In short, modern cosmology allows us to draw no conclusion about whether the universe has existed for a finite or infinite amount of time. And anyone who says differently is not being completely honest.

Next time: Why the BVG theorem is irrelevant to the Kalam Cosmological Argument!