What does it mean to look back in time? When astronomical observations are made of galaxies which are several billion light years distant, what can be said about where, as well as into what time the observations are looking?  How do such observations relate to the Hubble expansion of the universe?

The following is a mathmatical treatment of the factors involved.  It does not require more mathematics than undergraduate calculus.

Begin by thinking of the physical universe as the three dimensional hypersurface of an expanding four dimensional hypersphere. That is to say, assume the universe is a three dimensional "surface" within a four dimensional spatial framework.  No assumption about time as a fourth (or fifth) dimension is needed, although the math does encompass minimal assumptions beyond Newtonian/Euclidean cosmology, that is to say the R word.

If we slice this hypersphere with a plane, a two dimensional slice through this coordinate system, the three dimensional hypersurface becomes just an expanding circle of radius, r.

If you prefer, just start here and begin with the idea that this expanding circle is a toy model of a one dimensional universe embedded in two dimensional space so that it is closed on itself to form a circle.

Assume r = K*t to reflect the fact that we assume the universe is expanding at a constant rate K.

To account for distance between two objects within this one dimensional circular universe it is convenient to think of the distance between two points on the circle as an angle, formed by between the two lines from the center of the circle to the two points.  This is just a simple, convenient measure of distance.

Now let theta represent the angular measure of distance that light will
cover in time t.

Since the incremental distance covered in time dt is c*dt, it follows that r*d_theta = c*dt
or that d_theta = (c/K) dr/r

Now we integrate this expression to find the angular measure of the distance which light must cover while our toy universe expands from radius r_0 to radius r .

theta = c/K log(r/r_0) or the other way around:
r = r_0 exp((K/c)*theta)
This equation describes the path of a light signal as it leaves the source and moves through the universe at the speed of light.  Close to the source, it corresponds to the light cones usually used to describe relatavistic behavior.  In this curved universe, the sides of the cone are no longer straight lines but curved.

Now ask, how rapidly does the distance s between the end points of such a light path change as a function of the expanding universe?

We learn this by integrating r*d_theta:
ds = r*d_theta = r_0*exp((K/c) e)*d_theta
which integrates to become s = ((r_0*(c/K)*(1 - exp((K/c)*theta))

Now consider how this length s would vary in time as a function of the expansion of the hypersphere.

ds / dt = d(r_0*(c/K)*(1 - exp((K/c)*theta)))/dt
ds/ dt = K*(c/K)*(1 - exp((K/c)*theta))

Let v represent the speed at which the end points are separating:
v = c*(1 - exp(-(K/c)*e))

Since
This can be converted to a more convenient form:
v = c*(1 - (r/r_0))
v = c*(r_0 - r)/r_0 = c(T_0 - T_s)/T_0
v = c*a / T_0

We have now obtained an expression involving
1) recession velocity v
2) speed of light, c
3) distance from the reference point (eg, the earth)
4) age of the universe, T_0

H = v/c*a = 1/T_0

v/c*a corresponds to Hubble's constant. T_0 is the age of the universe.Hubble's constant provides information about the age of this universe,however K, our initially assumed rate of expansion of the circle has disappeared from the math.

Using Hubble's constant as 77 km/sec/m parsecs or 2.6E-18 km/sec/km gives an age for this model universe of 12.7 billion years. I expected K (equation 1) to show up in the answer. It doesn't. Isn't that odd? That seems to say that Hubble's constant is actually independent of the rate of expansion of the universe in this model. K could be anything. It could be larger than c. It could even be imaginary.