Problem: Assume (don't question) you know for certain that $$F=ma\mu(\frac{a}{a_0}),$$
where \(\mu\) is an unknown function. (I know how you feel, but put those feelings aside and think about it as a solely math problem😅)
Your goal is to determine the unknown function \(\mu\).
Another assumption is that \(a_0\) is found by the following:
In an expanding Universe $$ds^2=-c^2 dt^2 + R^2(t) dx^2,$$
(note that as a special case, for \(R(t)=1\) this yields the Minkowski metric)
Then \(a_0\) is just the acceleration of null rays of the above metric, meaning that if you let \(ds=0\), then $$cdt=\pm R(t) dx$$ so $$\frac{dx}{dt}=\pm \frac{c}{R},$$ then $$a_0 := \frac{d^2x}{dt^2}\Big|_{t=0}.$$
The simplest approach to solving this is to recall that, if we define Force as $$F=m\frac{d^2 x}{d\tau^2},$$ where \(d\tau=\frac{ds}{c}\), by chain rule we have $$\frac{d^2 x}{d\tau^2}=\frac{d^2 x}{dt^2}(\frac{dt}{d\tau})^2 + \frac{dx}{dt}\frac{d^2 t}{d\tau^2},$$
So by comparison with the assumption $$\mu=(\frac{dt}{d\tau})^2,$$
but this shows the problem: If we use the metric above, \(\frac{dt}{d\tau}\) is not a function of acceleration.
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