Naive Calculation of the Age of the Universe

[jeffbarrington]

I have marked this as high school level, although I am studying an undergraduate general relativity course, and just want to get some basics right.

Whenever I look for a 'cheap and dirty' method of calculation for the age of the universe, with a Hubble constant not changing with time, I am met with:

t = d/v

where I am meant to believe that t is the age of the universe, d is the separation between two galaxies at some great distance apart, and v is their current speed of separation, given by v = H_0d. However, they haven't always been moving at this speed apart from each other; up until now, they have been traveling at a speed < H_0d. Why are these 'derivations' failing to point out this flaw and what is the workaround?

Thanks

 

[Bandersnatch]

As you note, this derivation of the age of the universe is naive. That's all there is to it. As you correctly surmise, the naivety in it concerns the recessional velocities being constant. That is to say, the age it nets you is the age you'd get in an universe without dark energy accelerating the expansion and without matter and radiation retarding it. (This is called Milne expansion, btw.)

However, they haven't always been moving at this speed apart from each other; up until now, they have been traveling at a speed < H_0d.

This bit is incorrect. For most of the history of the universe, comoving observers have been receding from each other at recessional velocities higher than $H_0d$. It's only since approx. 5Gly ago that the velocities are increasing. See the graph below:

As you can see, the initial impulse was very high. During the period of high density, recessional velocities were retarded at a very high rate. Around the 8Gly point, due to continuous dilution of matter and radiation dark energy density started dominating, and recessional velocities begun increasing.

 

[Jorrie]

Just to ensure that Jeff understands the meaning of the graph, V_gen stands for the recession rate of a generic galaxy that presently has a recession rate of c, i.e. some galaxy that is somewhere on the Hubble sphere now, at a comoving distance of 14.4 billion light years from us. Due to the accelerated expansion, that galaxy is presently moving through our Hubble sphere from the 'inside'. It came into our 'then' Hubble sphere at t ~ 3.5 billion years.

 

[haushofer]

It's similar to calculating a time t=d/v for a car moving at a varying speed v(t). The proper way is to rephrase the expression into an integral.