|Numbers speak. [Img Src]|
Here is one of these non-problems. Did you know that the universe is spatially almost flat? There is a number in the cosmological concordance model called the “curvature parameter” that, according to current observation, has a value of 0.000 plus-minus 0.005.
Why is that a problem? I don’t know. But here is the story that cosmologists tell.
From the equations of General Relativity you can calculate the dynamics of the universe. This means you get relations between the values of observable quantities today and the values they must have had in the early universe.
The contribution of curvature to the dynamics, it turns out, increases relative to that of matter and radiation as the universe expands. This means for the curvature-parameter to be smaller than 0.005 today, it must have been smaller than 10-60 or so briefly after the Big Bang.
That, so the story goes, is bad, because where would you get such a small number from?
Well, let me ask in return, where do we get any number from anyway? Why is 10-60 any worse than, say, 1.778, or exp(67π)?
That the curvature must have had a small value in the early universe is called the “flatness problem,” and since it’s on Wikipedia it’s officially more real than me. And it’s an important problem. It’s important because it justifies the many attempts to solve it.
The presently most popular solution to the flatness problem is inflation – a rapid period of expansion briefly after the Big Bang. Because inflation decreases the relevance of curvature contributions dramatically – by something like 200 orders of magnitude or so – you no longer have to start with some tiny value. Instead, if you start with any curvature parameter smaller than 10197, the value today will be compatible with observation.
Ah, you might say, but clearly there are more numbers smaller than 10197 than there are numbers smaller than 10-60, so isn’t that an improvement?
Unfortunately, no. There are infinitely many numbers in both cases. Besides that, it’s totally irrelevant. Whatever the curvature parameter, the probability to get that specific number is zero regardless of its value. So the argument is bunk. Logical mush. Plainly wrong. Why do I keep hearing it?
Worse, if you want to pick parameters for our theories according to a uniform probability distribution on the real axis, then all parameters would come out infinitely large with probability one. Sucks. Also, doesn’t describe observations*.
And there is another problem with that argument, namely, what probability distribution are we even talking about? Where did it come from? Certainly not from General Relativity because a theory can’t predict a distribution on its own theory space. More logical mush.
If you have trouble seeing the trouble, let me ask the question differently. Suppose we’d manage to measure the curvature parameter today to a precision of 60 digits after the point. Yeah, it’s not going to happen, but bear with me. Now you’d have to explain all these 60 digits – but that is as fine-tuned as a zero followed by 60 zeroes would have been!
Here is a different example for this idiocy. High energy physicists think it’s a problem that the mass of the Higgs is 15 orders of magnitude smaller than the Planck mass because that means you’d need two constants to cancel each other for 15 digits. That’s supposedly unlikely, but please don’t ask anyone according to which probability distribution it’s unlikely. Because they can’t answer that question. Indeed, depending on character, they’ll either walk off or talk down to you. Guess how I know.
Now consider for a moment that the mass of the Higgs was actually about as large as the Planck mass. To be precise, let’s say it’s 1.1370982612166126 times the Planck mass. Now you’d again have to explain how you get exactly those 16 digits. But that is, according to current lore, not a finetuning problem. So, erm, what was the problem again?
The cosmological constant problem is another such confusion. If you don’t know how to calculate that constant – and we don’t, because we don’t have a theory for Planck scale physics – then it’s a free parameter. You go and measure it and that’s all there is to say about it.
And there are more numerological arguments in the foundations of physics, all of which are wrong, wrong, wrong for the same reasons. The unification of the gauge couplings. The so-called WIMP-miracle (RIP). The strong CP problem. All these are numerical coincidence that supposedly need an explanation. But you can’t speak about coincidence without quantifying a probability!
Do my colleagues deliberately lie when they claim these coincidences are problems, or do they actually believe what they say? I’m not sure what’s worse, but suspect most of them actually believe it.
Many of my readers like jump to conclusions about my opinions. But you are not one of them. You and I, therefore, both know that I did not say that inflation is bunk. Rather I said that the most common arguments for inflation are bunk. There are good arguments for inflation, but that’s a different story and shall be told another time.
And since you are among the few who actually read what I wrote, you also understand I didn’t say the cosmological constant is not a problem. I just said its value isn’t the problem. What actually needs an explanation is why it doesn’t fluctuate. Which is what vacuum fluctuations should do, and what gives rise to what Niayesh called the cosmological non-constant problem.
Enlightened as you are, you would also never think I said we shouldn’t try to explain the value of some parameter. It is always good to look for better explanations for the assumption underlying current theories – where by “better” I mean either simpler or can explain more.
No, what draws my ire is that most of the explanations my colleagues put forward aren’t any better than just fixing a parameter through measurement – they are worse. The reason is the problem they are trying to solve – the smallness of some numbers – isn’t a problem. It’s merely a property they perceive as inelegant.
I therefore have a lot of sympathy for philosopher Tim Maudlin who recently complained that “attention to conceptual clarity (as opposed to calculational technique) is not part of the physics curriculum” which results in inevitable confusion – not to mention waste of time.
In response, a pseudoanonymous commenter remarked that a discussion between a physicist and a philosopher of physics is “like a debate between an experienced car mechanic and someone who has read (or perhaps skimmed) a book about cars.”
Trouble is, in the foundations of physics today most of the car mechanics are repairing cars that run just fine – and then bill you for it.
I am not opposed to using aesthetic arguments as research motivations. We all have to get our inspiration from somewhere. But I do think it’s bad science to pretend numerological arguments are anything more than appeals to beauty. That very small or very large numbers require an explanation is a belief – and it’s a belief that has become adapted by the vast majority of the community. That shouldn’t happen in any scientific discipline.
As a consequence, high energy physics and cosmology is now populated with people who don’t understand that finetuning arguments have no logical basis. The flatness “problem” is preached in textbooks. The naturalness “problem” is all over the literature. The cosmological constant “problem” is on every popular science page. And so the myths live on.
If you break down the numbers, it’s me against ten-thousand of the most intelligent people on the planet. Am I crazy? I surely am.
*Though that’s exactly what happens with bare values.