Given infinite inflation would all combinations of particles be likely to occur?

Question

My question is, in the eternal inflation model of the universe, some theories predict that random thermal fluctuations would be likely to form most combinations of matter over infinite time. This is where the Boltzmann Brain problem comes in, the idea that given infinite time, thermal fluctuations would cause every unlikely combination of matter including brains.

Here's Sean Carroll writing on the issue:

https://arxiv.org/pdf/1702.00850.pdf "The eventual empty universe is therefore dominated by massless particles – photons and gravitons, neither of which are likely to assemble into conscious observers, since their interactions are so weak that it is hard to imagine forming bound states other than black holes. Nevertheless, there can be collisions between rare high-energy photons or gravitons, which could pair-produce electrons and positrons, or protons and anti-protons, and so forth. The general tendency of such pairs would be to re-annihilate rather quickly, but occasionally the new particles will have enough momentum to travel far apart from each other. Sometimes (rarely, as should be henceforth understood) many such collisions will happen nearby, producing enough nearby matter to assemble itself into a macroscopic object such as a brain."

My question is, given the model Carroll is using here, would particles really keep forming new fluctuations forever?

Excuse my ignorance, but if we live in a flat universe, I assumed massless particles like photons keep travelling forever outwards so they eventually wouldn't be available to make new combinations of matter.

Also, supposing protons don't decay (and I know that's a big suppose), I would have thought any particles that fluctuate into protons stay as protons, so eventually you wouldn't get enough other types of particles to make up matter. And with ongoing inflation, I would have thought protons travel too slowly to reach other matter. To put it another way. they would travel slower than other matter is moving away from them, given the ongoing expansion of space.

I'm not so much concerned here with the philosophy around Boltzmann brains as the science of what fluctuations would actually occur given infinite time.

I'm coming from a philosophy background, not a physics background, so please be gentle with your answers!

Answer 1

My question is, given the model Carroll is using here, would particles really keep forming new fluctuations forever?

Excuse my ignorance, but if we live in a flat universe, I assumed massless particles like photons keep travelling forever outwards so they eventually wouldn't be available to make new combinations of matter.

These are two different questions. The latter is simple to address: there is no "traveling outwards" because the Universe does not have a (mathematical) boundary i.e. you cannot a define any direction as "outwards". Photons, gravitons etc keep traveling forever on what can be seen as a smoothly connected hypersurface - sort of like a balloon's surface, but 3+1-dimensional rather than 2-dimensional. The fact they keep traveling and traversing all of space does not impede their ability to produce particle-antiparticle pairs via some physical process.

The first question is non-trivial, but this is why Sean Carroll distinguishes between 3 different quantum fluctuations:

1. Vacuum fluctuations: quantum mechanical processes that deviate from a classical analogue
2. Measurement-induced fluctuations: variations between multiple iterations of the same process
3. Boltzmann fluctuations: microstate changes in statistical mechanics that result in a change in entropy

The first type may persist forever since they can exist for stationary states where no time evolution occurs. The second type can persist forever insofar as there are observers (not necessarily sentient) to make measurements. It is the third type that may not persist forever if we indeed end up with a stationary final thermal state, albeit the text offers more nuance to this.

The bottom line is that there is no reason to assume that the type of fluctuations Carroll mentions in the excerpt you mention will cease at any given time in the remote future.

Also, supposing protons don't decay (and I know that's a big suppose), I would have thought any particles that fluctuate into protons stay as protons, so eventually you wouldn't get enough other types of particles to make up matter.

Particles do not fluctuate directly into any random combination of particles, but via specific processes. Protons are composite particles made of quarks, hence for them to be produced from some combination of photon-related process, you'd have to produce other particle species prior to that. Starting with photons, virtual pair-creation from a single photon or photon-photon scattering are purely QED processes. Fermion-antifermion pairs imply both leptons such as electrons, as well as quarks. By the time you have had enough iterations of such processes to assemble all the necessary components and rare conditions (as mentioned in the excerpt by Carroll) to form a proton, you would already have numerous other particles (both composite and not), of which many just as likely (or even more so) than the eventual proton. A proton may be a very stable final state, but not so overwhelmingly likely that almost every photon in the Universe would eventually lead up to produce one.

And with ongoing inflation, I would have thought protons travel too slowly to reach other matter. To put it another way. they would travel slower than other matter is moving away from them, given the ongoing expansion of space.

How fast a proton is moving depends on its overall energy, which is not completely unlikely to be very high as to reach velocities close to $c$ (especially if the photons that you begin with have high energy as well). Moreover, there is no "race" going on; particles may "run away" from some other particles at some region of space due to being faster, but others could be incoming from some other direction.

As for your inquiry with regards to inflation, every particle is moving away from each other at the same rate due to cosmological expansion - it doesn't have to do with how fast the particle is. In fact, if you actually study de Sitter space closely, you'll realize that even a massless/lightlike particle will eventually lose all causal contact with the rest of the Universe in the remote/infinite future. That is of course an asymptotic "ending": for pretty much the entirety of the "history" of de Sitter space, there are always some particles in causal contact to interact.