I wonder if there is any major part of physics that has changed so fundamentally in the last thirty years as cosmology.

Back in 1987 cosmology was more or less the last module we were taught as part of the astrophysics degree. But what we were taught then seems like basic arithmetic compared to today’s differential calculus.

The depth of the change in perspective has been brought home to me by reading Max Tegmark‘s Our Mathematical Universe. Tegmark’s book has only just been published but is already out of date in the sense that he speculates that the imprint of gravitational waves in the cosmic microwave background will be a key piece of evidence to support theories of an inflationary cosmology – and we now know that such imprints have been found.

But what is really shocking is Tegmark’s – convincing – argument that what we already knew supports inflation and inflation means we must live in an infinite multiverse. Moreover he completely clobbers the idea that multiverses are unscientific speculation: the evidence, he says, is all around us and the theory fully falsifiable in Karl Popper’s sense.

The book is an easy read – it is semi-autobiographical – and I have made a lot of progress with it in just a couple of days. Certainly recommended.

Imagine the universe was infinite (along the “quilted multiverse” pattern – namely that it streched on and on and we could only see a part). That would imply, assuming that the “cosmological principle” that one bit of the universe looked like any other, applied, that there were an infinite number of hydrogen atoms out there.

So, why is the universe not of infinite density? Because surely Shroedinger’s Equation means that there is a finite probability that electrons could be in any given region of space? (Doesn’t it?)

For any given electron the probability in “most” regions of space is zero in any measurable sense. But if there are an infinite number of electrons then the probability at a given point that there is an electron there is infinite, isn’t it?

OK, I have obviously got something wrong here because nobody is dismissing the “quilted multiverse” idea so simply – but could someone explain what it is I have got wrong?

Update: Is this because space-time is a continuum and the number of electrons a countable infinity?

Aleph-null () is the order (size) of the set of countably infinite objects. The counting numbers are the obvious example: one can start from one and keep on going. But any infinite set where one can number the members has the order of . (There are other infinities – eg that of the continuum, which have a different size of infinity.)

It is the nature of that proportions of it are also infinite with the same order. So 1% of a set with the order is also of order . To understand why, think of the counting numbers. If we took a set of 1%, then the first member would be 1, the second 101, the third 201 and so on. It would seem this set is of the size of the counting numbers, but it is also the case that because the counting number set is infinite with order , the 1% set must also be infinite and have the same order. In other words, if paradoxically, the sets are in fact of the same order (size) – .

The problem for cosmologists comes when considering the whether we can use observations of our “universe” to point to the experimental provability of theories of an infinite number of universes – the multiverse.

The argument runs like this: we have a theory that predicts a multiverse. Such a theory also predicts that certain types of universe are more typical, perhaps much more typical than others. Applying the Copernican argument we would expect that we, bog-standard observers of the universe – nothing special in other words – are likely to be in one of those typical universes. If we were in a universe that was atypical it would weaken the case for the theory of the multiverse.

But what if there were an infinite number of universes in the multiverse? Then, no matter how atypical any particular universe was (as measured by the value of various physical constants) then there would be an infinite number of such a typical universes. It would hardly weaken the case of the multiverse theory if it turned out we were stuck inside one of these highly atypical universes: because there were an infinite number of them.

This “measure problem” is a big difficulty for cosmologists who, assuming we cannot build particle accelerators much bigger than the Large Hadron Collider, are stuck with only one other “experiment” to observe – the universe. If all results of that experiment are as likely as any other, it is quite difficult to draw conclusions.

Greene seems quite confident that the measure problem can be overcome. I am not qualified to pass judgement on that, though it is not going to stop me from saying it seems quite difficult to imagine how.