I am a first year student at varsity and I am new to Astronomy. A question came up in my tut asking about the Hubble constant and cosmological time. I couldn't answer it because I have not understood the relationship between the two.
What I've heard is that H is constant for space(although I don't understand what that implies), and is not for time. What does this all mean?!
Please help...and please keep things simple.
Space itself is expanding (by which I mean the distance between two points in space is increasing even if those two points initially aren't in motion relative to each other) and the Hubble constant (H) is a measure of how fast it is expanding. The Hubble constant is often given in units of "km/s/Mpc", or kilometers per second per megaparsec. This means that if H is 70 km/s/Mpc, for an object 1 megaparsec away (about 3 million light years) the expansion of space will make it seem like it is moving away from us at a speed of 70 km/s. Something 0.5 megaparsecs away would appear to be moving away from us at a speed of 35 km/s. To repeat, this is due to the expansion of space itself.
Since km and Mpc are both units of distance, once can convert km to Mpc or Mpc to km and end up with a Hubble constant that would have the units of 1/s. Because of this, 1/H has units of seconds and is a measure of time. This is, in fact, a measure of the cosmological time, or the time since the Big Bang. This measure of the cosmological time is often called the "Hubble time" and it gives the time it would take to "rewind" the universe backwards to the Big Bang assuming that the Hubble constant has been constant throughout the existence of the universe. This isn't the case so the Hubble time is more of an estimation than an accurate measure of the age of the universe.
Here is a website with a bit more complete explanation and mathematical calculation of the Hubble time.
Cosmology is based on Einstein's theory of General Relativity, in particular, the standard model of cosmology assumes that the universe is isotropic, and hence, spatially homogeneous. As a result, one can take a 4-D spacetime, and split it into a 3+1 "Space" + "Time", this splitting is not unique, but, the spatial homogeneity assumption in cosmology allows one to define a cosmic time, in which these 3-d spatial slices are "threading through" a time-like vector that is orthogonal to each space like surface. Now, in cosmology, one has an equation, called the Raychaudhuri equation, that is usually given as H'(\tau), which measures the rate of expansion of the universe. To be precise, it actually measures the rate of expansion between two nearby particles, but for cosmology, the Raychaudhuri equation measures the relative rate of expansion of the universe, so that is what the generic Hubble parameter is.
Your question about cosmological time is directly related to \tau. Because G.R. is coordinate invariant, one is free to choose a time coordinate. Related to the discussion above, one typically defines a Hubble time, \tau by the following relationship: dt/d tau = 1/H, where "t" is standard clock time, and \tau is the cosmological time.
This equation is the relationship between the Hubble parameter/"constant" and the cosmological time.
