Derivation of Gott's formula for size of observable universe

[Buzz Bloom]

I understand that Gott derived a formula for calculating the size of the observable universe, and the value of the diameter based on current obsrvations is 93 Gly. Can someone please show the mathematical derivation of Gott's formula, or give a reference to a source which shows this derivation?

 

[Bandersnatch]

I understand that Gott derived a formula for calculating the size of the observable universe

Unless you're talking about a different Gott than the J.R. Gott of Princeton, then I don't think it's fair to say he 'derived' anything of the sort, as the equations are something like a century old.

In fact, you're probably thinking about this paper of his:
'A map of the universe' J.R.Gott et al. 2005
In which he (among other things) uses the Friedmann equations (equations 4&5) and observational parameters for the Hubble constant, densities and curvature collected by WMAP to arrive at the observable universe proper radius by the equation 11.
That's not his invention, though. It's a standard treatment to be found in any cosmology textbook, but with then-fresh numbers to plug into the equations.

If that's all new to you, then you might find the recent series of PF Insights articles written by Jorrie to be of some use: 'Approximate LCDM Expansion in Simplified Math'. The distances and horizons are covered in parts 2 and 3. Although you'll probably want to read it all anyway.
The only downside of those, IMO, is that they use some non-standard units that might confuse you if you've read anything else and don't pay attention to what's being done.

 

[George Jones]

The cosmological horizon is given by

$$d_{\mathrm{horizon}} = c \int^{t_0}_0 \frac{dt}{a\left(t\right)},$$

where ##a\left(t\right)## is the scale factor of the universe. For a good approximation, see the end of

https://www.physicsforums.com/attachment.php?attachmentid=59214&d=1370171072

 

[Buzz Bloom]

Hi Bandersnatch:

My appologies. I need to take better notes on the internet pages I find that lead me astray. I cannot now find the page I was previously looking at that was discussing the observable universe size and connected that with "Gott's formula".

Thanks for the link to "A Map of the Universe". From a quick scan it seems to have what I was looking for

Hi George:

Thank you for posting the formula for D_horizon. It is a useful part of what I was looking for. The rest of what I was looking for is how this formula is derived, and the "A Map of the Universe" article seems to have that.

Thanks for both your posts,
Buzz

 

[Buzz Bloom]

Hi George:

I looked through your "Redshift-Distance Relationhship" you cited paper and found in Section 5 the derivation I wanted based on the FLRW metric simplified for radial motion along comoving cooridnates. With this understanding, I now have a new question about the horizon problem I will ask in a separate thread.

Thanks again for your post,
Buzz