# How to apply equations to figure out how Hubble's constant affects the speed of light?

For my thought experiment, I will create my very own infinite expanding universe similar to our own assuming current cosmological theories are correct, except for a couple things:

$$\text{(Hubble's constant) }H_0 = \mathrm{1 s^{-1}}$$ $$\text{ (speed of light) }c = \mathrm{100 m/s}$$

In my universe, Hubble's constant never changes from $t=0\to\infty$ and the current time is $t= \mathrm{1 s}$. Now there is an observer $O_1$ who fires a photon at an observer $O_2$ in which the observers are separated by a distance $$d=\mathrm{100m}$$

Due to the expansion of the universe, $O_1$ is moving relative to $O_2$ at the following velocity

$$v = H_0d=\mathrm{100 m/s}$$

Clearly the photon will never reach $O_2$ since the photons can only travel at the speed of light, and already $O_2$ is receding too fast, so the photon could never hope to catch up. I believe then the radius of the Hubble Sphere for my universe is $\mathrm{100 m}$

So, here's what I want to be able to figure out:

What is the largest value we could assign $d$ for which the photon would reach $O_2$, where $d$ is the proper distance (as described here) at the current time ($t=\mathrm{1 s}$) from the $O_1$ in the direction of $O_2$?

Also, how far apart were $O_1$ and $O_2$ at $t=\mathrm {0}$?

I am trying to figure it out with these modified constants so I could wrap my head around the math in the real world. If I'm not mistaken there must be some integral calculus going on to figure it out, but I am having difficulty figuring it out because (for one) it has been so long since I took calculus, and (for two) the equations I know about are all with respect to $d$. For example, $v(d) = H_0d = 1d = d$, but what to do from there - and how to account for the increasing spacial distances in the formula.

For example at $t = 2\mathrm{s}$ you might think that $O_2$ would be at $d=200\mathrm m$ since it was traveling at $100 \mathrm {m/s}$ and $\mathrm {1s}$ has ellapsed, so it must have moved $\mathrm {100m}$ from the original $d=\mathrm{100m}$.

However, $d$ should be the proper distance, and we know that a point at $d=200\mathrm m$ was moving away from $O_2$ at $100 \mathrm{m/s}$ at $t=1 \mathrm {s}$ and there is no way $O_2$ could reach that far because it is outside of $O_2$'s Hubble Sphere (not to mention that hypothetically $O_2$'s proper distance should not change with respect to time. To complicate it further, along the 100m trip during $t \to 2\mathrm s$, $O_2$ has increased velocity proportional to $H_0$.

My thought was that the expansion of space must have affect the speed of light, since theories are that distant galaxies can move away at speed greater than the speed of light in relation to the coordinates of an observer. This would then mean that the effective speed of light would decrease as the value of $H_0$ increases, is that right?

I know that in the real world the value for $H_0$ is so small ($2.3×10^{−18}\mathrm{s^{−1}}$) and the value for $c$ is so big ($3.0×10^8 \mathrm{m/s}$), there would need to be an insane 26 significant figures in any sort of measurement before the expansion of the universe affected the speed of light.

But still - hypothetically with my thought experiment it seems to have a non-zero effect - making me question what it actually means when someone tells you the speed of light is a particular value (like I did earlier by saying it was 100m/s) - would it mean the effective speed given the expansion of the universe, or the speed with relation to proper distances relative to a particular time (i.e. how far it is from where it was emitted, or how close it is to point it was aimed at)?

(also I just like Math and would like to know the math behind trying to figure out the answer to my experiment)

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