Does angular diameter distance vary with composition of the Universe?

[ck99]

I was just reading about how the redshift-relation for angular diameter distance is calculated, and the example in my textbook used a matter-dominated universe to calculate the formula. It seems to rely heavily on the relationships between t, a(t) and H(t), which are different in radiation dominated universe. Does the final result end up the same regardless of the content of the universe? I have tried to work through with different values but it's beyond my abilities!

 

[Mordred]

This is one of those cases where its better if you show your calculations. It will help us understand your level of knowledge.

https://www.physicsforums.com/showpost.php?p=3977517&postcount=3

the link above shows how the latex commands work to post math equations.

Also are you sure you don't want the proper distance or commoving distance? angular diameter distance correlates an objects actual size vs its apparent size due to distance.

 

[marcus]

I was just reading about how the redshift-relation for angular diameter distance is calculated, and the example in my textbook used a matter-dominated universe to calculate the formula. It seems to rely heavily on the relationships between t, a(t) and H(t), which are different in radiation dominated universe. Does the final result end up the same regardless of the content of the universe? I have tried to work through with different values but it's beyond my abilities!

Nice question. I would imagine it does depend on whether one assumes matter dominated or radiation dominated.
But isn't it true that we only use angular size distance in matter-dominated setting? You need some material object to have a size. Even if it is just a patch of overdensity or the wavelength of a pressure-wave in a gaseous medium.

Basically I'm going to duck your question but I want to offer you an article I really like, by Charles Hellaby at Capetown SA. It is about the MAXIMUM of the angular size distance. I think it is about 5.8 billion LY. And that maximum is related to the mass of the universe contained within that radius. So observationally determining the greatest angular size distance of any object we can see--the "farthest" object in that sense--would be a way of getting an observational handle on the mass of the U.

It is a curious fact that the samesize object looks bigger the farther out you go beyond 5.8 Gly (with Ned Wright calculator the redshift with the maximum AngSizeDist is around 1.64, you can check). So the maximum of that distance which it is possible to measure is 5.8 Gly and those are *fairly nearby galaxies*! they are currently only about 15 Gly from us!

Here's Hellaby's 2006 paper
http://arxiv.org/abs/astro-ph/0603637

Using Jorrie's calculator you can see the max in the DISTANCE THEN column, that is the galaxy's proper distance from us at the moment when it emitted the light that we are getting from it today.
Rows are indexed by stretch factor S = z+1 so the max D_then occurs around S=2.64 depending on which model parameters you use (Planck, WMAP, or Ned Wright's really out of date ones )

 

[Mordred]

Nice paper. Makes me wonder how that would apply when you use the angular diameter distance
as applied via a gravitational lens. Lol gives me something to look up

 

[marcus]

Here's a little table with Jorrie's, using the Planck model parameters that just came out last month.
It shows the maximum. It doesn't matter too much which model parameters. The result is roughly the same as Ned Wright's but you get a table. over whatever range you specify with however many steps. Here for simplicity I chose 10 steps going from S=10.9 down to the present-day S=1

$${\scriptsize \begin{array}{|c|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{∞} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline14.4&17.3&3400&67.92&0.693&0.307\\ \hline \end{array}}$$ $${\scriptsize \begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&a=1/S&T (Gy)&R (Gly)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)&a'R_{0}\\ \hline 10.900&0.092&0.479&0.720&31.375&2.878&4.391&1.366&1.834\\ \hline 8.584&0.116&0.686&1.030&29.384&3.423&5.344&1.967&1.628\\ \hline 6.760&0.148&0.982&1.472&27.142&4.015&6.454&2.829&1.447\\ \hline 5.324&0.188&1.404&2.098&24.621&4.625&7.722&4.066&1.289\\ \hline 4.192&0.239&2.005&2.980&21.797&5.199&9.132&5.836&1.153\\ \hline 3.302&0.303&2.855&4.200&18.648&5.648&10.642&8.365&1.038\\ \hline 2.600&0.385&4.044&5.833&15.177&5.837&12.179&11.957&0.949\\ \hline 2.048&0.488&5.676&7.891&11.427&5.581&13.633&17.014&0.891\\ \hline 1.612&0.620&7.837&10.232&7.510&4.658&14.882&24.034&0.873\\ \hline 1.270&0.788&10.560&12.525&3.620&2.851&15.835&33.583&0.905\\ \hline 1.000&1.000&13.787&14.400&0.000&0.000&16.472&46.279&1.000\\ \hline \end{array}}$$
Time now (at S=1) or present age in billion years:13.787
'T' in billion years (Gy) and 'D' in billion light years (Gly)

The version of Jorrie's I used here is:
http://www.einsteins-theory-of-relativity-4engineers.com/TabCosmo9p.html
that I happen to have in my signature.
You can see angular size distance attaining max around S=2.6 (subtract one to get redshift)