# A photon in expanding Universe (a snail on a tree)

I want to know how far a snail can reach in expanding universe. It has a constant speed c = 1 and tree is expanding at speed $v= H_0 D$, with Hubble constant $H_0 = 1$. Here D(T) is the distance of snail from the origin that I am looking for. I want to know how it depends on time T.

At every moment of time, t, D increases as $dD = H_0 D dt + c dt = (D + 1)dt$ because we add the tree growth to the snail speed. This is non-homogenous equation and I had to look into handbook to get the result $D(T) = e^T-1$. This looks right, as we have snail at the origin, D = 0, at time 0.

However, I do not know how to compute inhomogeneity myself. I keep in mind only that exponentials are solutions of homogenous equations. For instance, the Hubble equation, $dD = r_0 D\:dt$, gives exponentially expanding Universe $D = r_0 e^t$. To avoid inhomogeneity (and differential equations altogether), I tried to derive the solution right in the closed form.

I considered this way: in the first moment of time, snail travels $r_0 = dt$. This length starts expanding, according to Hubble law, $r(T-dt) = r_0 e^{T-dt} = e^{T-dt} dt$ to the snail's distance because it has $T-dt$ seconds to expand. This is the contribution of the first snail step to the resulting distance. During the next step, snail makes another $r_0 = dt$, but it has $dt$ less time to expand, resulting in contribution of $e^{T-2dt} dt$. The third step, will produce the same $r_0$, expanded to $e^{T-3dt} dt$ and so on, until, in the final step there will be no time to expand at all, $e^{T-T} dt = dt$.

So, we need to sum up the infinitecimal plots, whose sizes grow exponentially with t. In the form $D(T) = \int r(t) dt$, where $r(t) = e^{T-\int dt = T-t}$. Likewise $\int_0^Y {y dy} = \int_0^Y {(Y-y) dy}$, I think that we'll get the same result if, instead of reducing expansion time linearly, from T to 0, it would be increased linearly, from 0 to T, $D = \int_0^T e^{T-t} dt = \int_0^T e^{t} dt = e^t|_0^T = e^T-1$

Surprisingly, while I typed this text, the closed form integral matched the diffeq solution above! I had a discrepancy on the paper. It seems not a question anymore. Is it ok to leave the solution on this site? You may just check if it is ok.

So, photon with a constant speed, 1, in expanding Universe travels at $d(e^t-1)/dt = e^t$ -- exactly the speed of expanding Universe! Otherwise, you may answer why Ant on a rubber rope is consedered as a model of Universe expansion rather than my problem? I see that ant has it tree growing linearly, rather than exponintially, which cannot correspond to the Hubble Universe expansion.