Based on the smallest number of photons that can deliver information about a star, what is the largest size of the universe? I was watching a summary video about the Banach-Tarski paradox and it led me to a question.

If I (a) assume the universe is infinite and (b) assume it is possible to receive a single photon of light from any star within it that has a line-of-site to the Earth and (c) assume that the photon has travelled an infinite distance, then such a photon, if enough information could be extracted from it, would prove the universe is infinitely large.

Today we have finite observational and recording capabilities.  I believe (not being a physicist or astronomer myself) that we can't prove the universe is larger than some size, X, because we cannot detect lower than a particular number of photons, Y, to ascertain the distance they travelled.

So, assuming my assumptions aren't assinine (they very well could be), and ignoring our present observational limitations, what is the smallest number of photons from which we could discern information about the distance those photons travelled, and what distance would that represent?
So, assuming my assumptions aren't assinine (they very well could be) ...


Your words, not mine. Your "very well could be" is the case.



Assume the universe is infinite

The universe might well be infinitely large, but the observable universe is all that we can possibly see. By all signs, the universe is 13.8 billion years old. We can't see the stuff that is beyond the observable universe because not enough time has passed for light to have traveled that far.

To make matters worse, the early universe was very hot and dense. Photons traveled but a short distance before being absorbed. We can't see those very early photons because they no longer exist. This changed about 379,000 years after the universe formed, at which time the universe had sufficiently cooled so as to allow photons to roam free. The photons let loose by that clearing of the universe form the cosmic microwave background radiation.


Assume it is possible to receive a single photon of light from any star within it that has a line-of-site to the Earth

Per the big bang theory, the first stars formed about 100 to 200 million years after the formation of the universe. Stars that formed at that time beyond the observable universe: We can't see them. Due to the continuing (and possibly accelerating; google "dark energy") expansion of the universe, we will never see those stars.

There's a famous paradox related to your question: Why aren't we bathed night and day by light from an infinite number of stars if the universe is uniform and is infinitely old> This is Olber's paradox. The solution is simple: The universe is not infinitely old.


Assume that the photon has travelled an infinite distance, then such a photon, if enough information could be extracted from it, would prove the universe is infinitely large

The only information that can be extracted from a single photon is its frequency, and even that is a bit fuzzy due to quantum mechanics. A photon from a remote star is indistinguishable from a photon of the same frequency emitted a nearby lightbulb.