Why is a statistical explanation for Baryon Asymmetry wrong?

[Buzz Bloom]

I apologize if this is not the correct forum for this thread.

I have tried to find a discussion regarding this question on the Internet without any success. The Wikipedia discussion

https://en.wikipedia.org/wiki/Baryon_asymmetry​

makes no mention of any statistical explanation, so I understand that the idea must be flawed, but I do not understand in what way it is flawed.

The following is my awkward attempt to make a clear statement about this idea.

At time T₁, the universe had an equilibrium in the creation of Baryon Anti-Baryon Pairs (BABPs), and in their mutual annihilation. After T₁, there were more annihilations than creations. Imagine a very large sphere, say of radius R, with a center, using co-moving coordinates, at the same point where the sun is now. Also, assume R increases with time in proportion to the scale factor a. (BTW, I am not sure what value of R would be best for illustrating this “statistical explanation” idea.) That is, since the time T₁ the universe is dominated by radiation, until matter began to significantly affect the function a=f(t),

R = m * T^1/2,​

where m is the constant of proportionality.

Near the outside of the boundary, when most BABPs are created one of the pair will cross the boundary and the other not. (I am not sure what a reasonable fraction of R might be appropriate as being “near” the boundary.) The same is true inside the boundary. For each particle that crosses the boundary, in either direction, it either adds or subtracts one either to or from the baryon number inside the boundary. For a period before T₁, BABPs crossed the boundary, and for a period after T₁ more crossed the boundary until no more BABPs were created, say at the time T₂. Let T₃ be the time when all BABPs have been annihilated. Let us also assume the value of the scale factor a = 1 for

T = T₃.​

Let N be the total number of BABPs which crossed the boundary prior to T₃. The total change in the baryon number inside the sphere is then statistically a Gaussian distribution with a mean

m = 0,​

and a standard deviation of

σ = 0.5 sqrt(N).​

Why is it not plausible that a random number from this distribution represents the current baryon number within a sphere centered on the sun with a radius

R* = a R,​

where a is the current value of the scale factor?

I am aware of some possible issues with this idea, but my math skills are not sufficient for me to quantify these issues.

(1) At time T₃, the Baryon Number Density per unit volume (BND) for different regions of the sphere will be different. There is an issue about whether this variability within the sphere (with its radius growing proportionately with a) would become sufficiently close to uniform by the present time so that this variability would not noteworthy.

(2) In order for (1) to be so, the baryons between time T₃ and now would have to interact with each other sufficiently so that BND becomes much less variable. This requires that baryons move a significant distance across the sphere. However, as they move the will interact with other baryons which would change the direction of their movement. Therefore their path would be random walks, and the distance traveled between T₃ and now would be proportional to the square root of the number of interactions along their path. This might very much limit the amount of mixing that would take place during T₃ to now, and so to perhaps become insufficient to make (1) valid.

All comments would be appreciated. In particular, I would very much like to see any citations of references about this idea having being analyzed to demonstrate mathematically why it fails to be plausible.

 

[PeterDonis]

Any explanation of this kind for baryon asymmetry needs to be able to account for the fact that our observable universe contains no appreciable antibaryons, but approximately $10^{80}$ baryons. So the size $R$ needs to be roughly the size of our observable universe at the time when baryon-antibaryon annihilation was proceeding, and the number $N$ in your formulas needs to be large enough to make it plausible that $0.5 \sqrt{N} \approx. 10^{80}$. But we know that the ratio of photons to baryons is about a billion to one, which means that the total number of baryons before annihilation in our observable universe, which should be at least roughly of the same order of magnitude as your $N$ (the total number of baryons crossing the boundary of the sphere with radius $R$), was roughly $10^{89}$. Which is, of course, about 71 orders of magnitude smaller than the $N$ that would be required for the standard deviation formula I just gave. That, in a nutshell, is what I see as the fundamental problem with this hypothesis.

 

[Buzz Bloom]

But we know that the ratio of photons to baryons is about a billion to one, which means that the total number of baryons before annihilation in our observable universe, which should be at least roughly of the same order of magnitude as your NN (the total number of baryons crossing the boundary of the sphere with radius RR), was roughly 108910^{89}.

Hi Peter:

Thank you for your very clear (as usual) and useful answer to my question.

Regards,
Buzz