I have a question regarding the most distant observeable objects in the universe, how do we know their distance from us as light and radio waves from them travel through an exponentially expanding space with time?
We can see galaxies that are, and always have been, apparently receding from us faster than light. We can see them because the rate of expansion was decelerating for billions of years, so their light eventually found itself in places that weren't apparently receding from us faster than light. In an expanding universe, the distance where an object is receding at c is known as the Hubble distance and it marks the radius of a conceptual sphere around us known as the Hubble sphere.
The galaxies that were receding at the speed of light when they emitted the light we are now seeing have a redshift of around z=1.4 which equates to a light travel time of around 9.1 billion years. These galaxies were around 5.7 billion light years away from us when they emitted the light we see (angular diameter distance), but their light took over 9 billion years to reach due to the expansion of the universe. Those galaxies are now estimated to be something around 14 billion light years away (comoving distance).
But there is a distance at which we will never be able to see an event, due to the accelerating rate of expansion. It is known as the cosmological event horizon and is currently estimated to be around 16 billion light years away. In simplistic terms, it is the distance at which the light from an event will never make it to the edge of our Hubble sphere, due to the accelerating rate of expansion.
So right now, the edge of our Hubble sphere is around 14 billion light-years away and the Cosmological Event Horizon is around 16 billion light years away. We will never be able to see events that are happening now if they are happening more than 16 billion light years away.
If the rate of expansion were decelerating now, we would have our Hubble radius moving outward bringing the light from more distant galaxies into our eventual view. But as the rate seems to be accelerating, this means that galaxies will move out past our cosmological event horizon so as time goes on we will end up seeing fewer and fewer galaxies until one day in the dim and distant future we will only be able to see the galaxies that are part of our local group.
Have a look at this PDF article, it is an excellent introduction to the subject.