I totally mean it: Inflation never solved the flatness problem.

I’ve had many interesting reactions to my recent post about inflation, this idea that the early universe expanded exponentially and thereby flattened and smoothed itself. The maybe most interesting response to my pointing out that inflation doesn’t solve the problems it was invented to solve is a flabbergasted: “But everyone else says it does.”

Not like I don’t know that. But, yes, most people who work on inflation don’t even get the basics right.

Inflation flattens the universe like
photoshop flattens wrinkles. Impressive!
[Img Src]

I’m not sure why that is so. Those who I personally speak with pretty quickly agree that what I say is correct. The math isn’t all that difficult and the situation pretty clar. The puzzle is, why then do so many of them tell a story that is nonsense? And why do they keep teaching it to students, print it in textbooks, and repeat it in popular science books?

I am fascinated by this for the same reason I’m fascinated by the widely-spread and yet utterly wrong idea that the Bullet-cluster rules out modified gravity. As I explained in an earlier blogpost, it doesn’t. Never did. The Bullet-cluster can be explained just fine with modified gravity. It’s difficult to explain with particle dark matter. But, eh, just the other day I met a postdoc who told me the Bullet-cluster rules out modified gravity. Did he ever look at the literature? No.

One reason these stories survive – despite my best efforts to the contrary – is certainly that they are simple and sound superficially plausible. But it doesn’t take much to tear them down. And that it’s so simple to pull away the carpet under what motivates research of thousands of people makes me very distrustful of my colleagues.

Let us return to the claim that inflation solves the flatness problem. Concretely, the problem is that in cosmology there’s a dynamical variable (ie, one that depends on time), called the curvature density parameter. It’s by construction dimensionless (doesn’t have units) and its value today is smaller than 0.1 or so. The exact digits don’t matter all that much.

What’s important is that this variable increases in value over time, meaning it must have been smaller in the past. Indeed, if you roll it back to the Planck epoch or so, it must have been something like 10^-60, take or give some orders of magnitude. That’s what they call the flatness problem.

Now you may wonder, what’s problematic about this. How is it surprising that the value of something which increases in time was smaller in the past? It’s an initial value that’s constrained by observation and that’s really all there is to say about it.

It’s here where things get interesting: The reason that cosmologists believe it’s a problem is that they think a likely value for the curvature density at early times should have been close to 1. Not exactly one, but not much smaller and not much larger. Why? I have no idea.

Each time I explain this obsession with numbers close to 1 to someone who is not a physicist, they stare at me like I just showed off my tin foil hat. But, yeah, that’s what they preach down here. Numbers close to 1 are good. Small or large numbers are bad. Therefore, cosmologists and high-energy physicists believe that numbers close to 1 are more likely initial conditions. It’s like a bizarre cult that you’re not allowed to question.

But if you take away one thing from this blogpost it’s that whenever someone talks about likelihood or probability you should ask “What’s the probability distribution and where does it come from?”

The probability distribution is what you need to define just how likely each possible outcome is. For a fair dice, for example, it’s 1/6 for each outcome. For a not-so-fair dice it could be any combination of numbers, so long as the probabilities all add to 1. There are infinitely many probability distributions and without defining one it is not clear what “likely” means.

If you ask physicists, you will quickly notice that neither for inflation nor for theories beyond the standard model does anyone have a probability distribution or ever even mentions a probability distribution for the supposedly likely values.

How does it matter?

The theories that we currently have work with differential equations and inflation is no exception. But the systems that we observe are not described by the differential equations themselves, they are described by solutions to the equation. To select the right solution, we need an initial condition (or several, depending on the type of equation). You know the drill from Newton’s law: You have an equation, but you only can tell where the arrow will fly if you also know the arrow’s starting position and velocity.

The initial conditions are either designed by the experimenter or inferred from observation. Either way, they’re not predictions. They can not be predicted. That would be a logical absurdity. You can’t use a differential equation to predict its own initial conditions. If you want to speak about the probability of initial conditions you need another theory.

What happens if you ignore this and go with the belief that the likely initial value for the curvature density should be about 1? Well, then you do have a problem indeed, because that’s incompatible with data to a high level of significance.

Inflation then “solves” this supposed problem by taking the initial value and shrinking it by, I dunno, 100 or so orders of magnitude. This has the consequence that if you start with something of order 1 and add inflation, the result today is compatible with observation. But of course if you start with some very large value, say 10⁶⁰, then the result will still be incompatible with data. That is, you really need the assumption that the initial values are likely to be of order 1. Or, to put it differently, you are not allowed to ask why the initial value was not larger than some other number.

This fineprint, that there are still initial values incompatible with data, often gets lost. A typical example is what Jim Baggot writes in his book “Origins” about inflation:

“when inflation was done, flat spacetime was the only result.”

Well, that’s wrong. I checked with Jim and he totally knows the math. It’s not like he doesn’t understand it. He just oversimplifies it maybe a little too much.

But it’s unfair to pick on Jim because this oversimplification is so common. Ethan Siegel, for example, is another offender. He writes:

“if the Universe had any intrinsic curvature to it, it was stretched by inflation to be indistinguishable from “flat” today.”

That’s wrong too. It is not the case for “any” intrinsic curvature that the outcome will be almost flat. It’s correct only for initial values smaller than something. He too, after some back and forth, agreed with me. Will he change his narrative? We will see.

You might say then, but doesn’t inflation at least greatly improve the situation? Isn’t it better because it explains there are more values compatible with observation? No. Because you have to pay a price for this “explanation:” You have to introduce a new field and a potential for that field and then a way to get rid of this field once it’s done its duty.

I am pretty sure if you’d make a Bayesian estimate to quantify the complexity of these assumptions, then inflation would turn out to be more complicated than just picking some initial parameter. Is there really any simpler assumption than just some number?

Some people have accused me of not understanding that science is about explaining things. But I do not say we should not try to find better explanations. I say that inflation is not a better explanation for the present almost-flatness of the universe than just saying the initial value was small.

Shrinking the value of some number by pulling exponential factors out of thin air is not a particularly impressive gimmick. And if you invent exponential factors already, why not put them into the probability distribution instead?

Let me give you an example for why the distinction matters. Suppose you just hatched from an egg and don’t know anything about astrophysics. You brush off a loose feather and look at our solar system for the first time. You notice immediately that the planetary orbits almost lie in the same plane.

Now, if you assume a uniform probability distribution for the initial values of the orbits, that’s an incredibly unlikely thing to happen. You would think, well, that needs explaining. Wouldn’t you?

The inflationary approach to solving this problem would be to say the orbits started with random values but then some so-far unobserved field pulled them all into the same plane. Then the field decayed so we can’t measure it. “Problem solved!” you yell and wait for the Nobel Prize.

But the right explanation is that due to the way the solar system formed, the initial values are likely to lie in a plane to begin with! You got the initial probability distribution wrong. There’s no fancy new field.

In the case of the solar system you could learn to distinguish dynamics from initial conditions by observing more solar systems. You’d find that aligned orbits are the rule not the exception. You’d then conclude that you should look for a mechanism that explains the initial probability distribution and not a dynamical mechanism to change the uniform distribution later.

In the case of inflation, unfortunately, we can’t do such an observation since this would require measuring the initial value of the curvature density in other universes.

While I am at it, it’s interesting to note that the erroneous argument against the heliocentric solar system, that the stars would have to be “unnaturally” far away, was based on the same mistake that the just-hatched chick made. Astronomers back then implicitly assumed a probability distribution for distances between stellar objects that was just wrong. (And, yes, I know they also wrongly estimated the size of the stars.)

In the hope that you’re still with me, let me emphasize that nevertheless I think inflation is a good theory. Even though it does not solve the flatness problem (or monopole problem or horizon problem) it explains certain correlations in the cosmic-microwave-background. (ET anticorrelations for certain scales, shown in the figure below.)

Figure 3.9 from Daniel Baumann’s highly recommendable lecture notes.

In the case of these correlations, adding inflation greatly simplifies the initial condition that gives rise to the observation. I am not aware that someone actually has quantified this simplification but I’m sure it could be done (and it should be done). Therefore, inflation actually is the better explanation. For the curvature, however, that isn’t so because replacing one number with another number times some exponential factor doesn’t explain anything.

I hope that suffices to convince you that it’s not me who is nuts.

I have a lot of sympathy for the need to sometimes oversimplify scientific explanations to make them accessible to non-experts. I really do. But the narrative that inflation solves the flatness problem can be found even in papers and textbooks. In fact, you can find it in the above-mentioned lecture notes! It’s about time this myth vanishes from the academic literature.