Can distant galaxies move away from us faster than the speed of light?

If we define the speed of a distant galaxy relative to us as \[ v = \dot{a}(t) r, \] where \(a(t)\) is the scale factor, and \(r\) is the comoving coordinate of the galaxy relative to us, then for sure, with large enough \(r\), \(v\) will be greater than the light speed \(c\).

This does not violate special relativity, because special relativity only applies to a local flat patch of the space-time. The global space-time is made of such small locally flat patches.

In the following we choose a unit system such that \(c=1\), and define \(r_\text{c}{}_{}\) such that \(\dot{a} r_\text{c}{}_{} = 1\).

One question: for a distant galaxy with an apparent speed greater than \(c\) at cosmological time \(t\), is it possible that a photon emitted at \(t\) by that galaxy can still reach us?

This basically amounts to calculation of future horizon. Assuming spatial curvature is zero, we have \[ \int_t^\infty \frac{dt'}{a(t')} = r_\text{hor}{}_{}, \] where \(r_\text{hor}{}_{}\) is the horizon radius. The question becomes whether \(r_\text{hor}{}_{} > r_\text{c}{}_{}\). If this inequality holds, then a photon emitted by the galaxy moving away from us faster than the speed light can reach us in finite time.

Let \(a(t)\propto t^\alpha\), we have \[ \int_t^\infty \frac{dt'}{a(t')} = \frac{1}{1-\alpha} t'^{1-\alpha}|^{\infty}_{t}{}_{}. \] For a matter-dominated universe, \(\alpha = 2/3\), hence \(r_\text{hor}{}_{} = \infty\), meaning that no matter how far the galaxy is, eventually the photons emitted by it will reach us in finite time --- even if its current apparent speed is faster than light.

What about accelerated expansion? Let \(a(t) = a_0 e^{H t}\), then \[ r_\text{hor}{}_{} = \int_t^\infty \frac{dt'}{a(t')} = \frac{1}{H a_0} e^{-H t}, \] and \[ r_\text{c}{}_{} = \frac{1}{H a_0} e^{-H t}. \] The two are exactly the same. Hence for such a universe, photons emitted by galaxies moving away from as with speed higher than the light speed will not reach us no matter how long we wait.