Relationship between the age of the Universe and the Hubble horizon

[Anthony111]

i got a bit lost in the responses to my last question so I am guessing this one is really going to be beyond me.

Assumptions I have used for my questions are:

· Speed of light = 299792.458 km/s

· Hubble constant = 71 km/s/Mpc (I know about the tension of H0 being 68 and 73 but don't think it alters my question)

· 1 Mpc = 30,856,776,000,000,000,000 km

· 1 Mpc = 3,261,563.7967311 ly

· 1ly = 365.25 days1. At a Hubble constant of 71km/s/Mpc it would take space 13,771,721,514.72 years to contract from a from 1 Mpc in distance.

30,856,776,000,000,000,000 km / 71 km/s = 434,602,478,873,239,436.619718 seconds

434,602,478,873,239,436.619718 seconds / 60 = 7.243,374,647,887,323.943661 minutes

7.243,374,647,887,323.943661 minutes / 60 = 120,722,910,122.085727 hours

120,722,910,122.085727 hours /24 = 5,030,121,283,255.086071 days

5,030,121,283,255.086071 days / 365.25 = 13,771,721,514.72 years

As I understand it, this is how we approximate the age of the universe. (I say approximate as H0 has decayed over time)2. At a Hubble constant of 71km/s/Mpc the rate of the expansion of space would take 13,771,721,514.72 ly to be traveling at c away from a given point.

299792.458 km/s / 71 km/s = 4222.428985 Mpc

4222.428985 Mpc * 3,261,563.796731 ly = 13,771,721,514.72 ly

This is simply telling us how far from a single point it would take the expansion of space to be expanding at c. It seems to be a different calculation and yet the answer is the same other than the first one is years in time and the second one is light years in distance.

Why is it that the age of the universe seems to be equal to the expansion of space reaching light speed?

 

[Bandersnatch]

When you look at the Hubble constant, you should notice that its dimensions are [distance]/[time]/[distance]. So it's actually equivalent to 1/[time].
This simply means that H tells you that if you pick any distance whatsoever, it will have grown to its current value in some characteristic time. By working out the units you end up with something like 1/144% per million years, or indeed 100% per ~13.8 billion years (under the assumption H stays constant in time). That's the time needed for any distance to grow from 0 to its current value.
It follows that a point that is farther away needs higher recessional velocity to cover its higher distance in the same time (that's just Hubble's law).
So if you now look at how H0 is expressed in the traditional units, i.e. ~70 km/s/Mpc, it tells you that a point that is 1 Mpc away needs this particular velocity of ~70 km/s to cover the distance of 1 Mpc in 13.8 Gyr. 2 Mpc would require 140 km/s, and so on.
Now, it should be unsurprising that if instead of 1 or 2 Mpc we pick a distance that is equal to 13.8 Gyr times the speed of light (13.8 Glyr), the speed required to cover this distance in 13.8 Gyr is going to be c.