# Apparent size of objects at cosmological distances

1. Jan 3, 2015

### krater

Maybe this is a complicated question but something I once read on another forum long ago has been rolling through my head lately. In discussing the observational size of distant objects (GLy) a comment was made to the effect that as distance increases an object's size seems to reach a lower limit and then start to expand radially with further distance.

Sounds counterintuitive, but the way I wrapped my head around it seemed to make sense as when one thinks about it there exists an obviously fixed area of sky; an observer from a static earth would therefore see light radiated from an object 500 million light years away with a diameter larger than light radiated at the same time as the "first" light reaches the observer, which assuming a rate of expansion faster than light could be far greater than double the original separation distance.

Taken to far distant lengths, galaxy which radiated its light our way many billions of years ago would appear anomolously large for its true distance by virtue of its light having left on its travels so long ago that the object's current distance (calculated by redshift ect) would far exceed its possible value not accounting for cosmological expansion.

Anyway this is all a bit disjointed I know but does any of this make sense to those with education? Can anyone divine out the "real" question I have with what I can come up with here? Thanks all.

2. Jan 4, 2015

### marcus

That's right, I'll look up the proper distance at which things look smallest. I think it is around redshift z=1.6.
You might be able to get something out of the "Lineweaver Figure 1" link in my signature. the top panel shows a teardrop shape lightcone . That figure is plotted using the "proper" distance, the real distance the object was if you could have paused expansion at that moment to give time to measure.

The point is that when the light starts out towards us, it at first is dragged back by expansion of distance, and actually loses ground. The widest spot on the bulge of the lightcone is where it begins to make headway towards us.

The angular size of an object depends on how far it was from us when it emitted the light that is now reaching us. Some things that emitted their light a very long time ago and are NOW a very great distance from us were nevertheless quite CLOSE when they emitted the light, so they make a corresponding large angle in the sky, "look big", compared with other objects the same real size.

Last edited: Jan 4, 2015
3. Jan 4, 2015

### marcus

Yes, the widest girth of the lightcone is at redshift z=1.6. corresponds to a "distance THEN" (i.e. at time of emission) of about 5.8 billion LY.
Those would be the objects that look the smallest, because Dthen is a maximum. In the table it shows up as 5.837. Do you see it?
The light was emitted in about year 4 billion. Distances have expanded since then by a factor of about S=2.6. The "scale factor" a, which they use to keep track of the size of distances relative to the present size a=1, was 1/2.6 back then, you can see it on the table, lefthand column. The redshift, which is defined as S-1, is of course 1.6.

Light which has been traveling longer, and is more redshifted than 1.6, comes from something that has to look bigger because it was NEARER when it emitted the light. E.g. look at the top row of the table. A redshift z=4 object was only 4.87 billion LY from here when it emitted the light. So for the same real size it has to look bigger than a z=1.6 objet, which was 5.83 billion LY when it emitted the light.

You can make tables with your own choice of S limits and number of steps if you want. The table making utility is called "Lightcone". A PF member named Jorrie created it. It will go back farther in time than what I showed here. And it will also tabulate the future stages of expansion. Also can plot curves. You might want to check it out. Link is in my signature
http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html
$${\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$$ $${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&z&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly) \\ \hline 0.200&5.000&4.000&1.5417&2.3019&23.910&4.782\\ \hline 0.217&4.613&3.613&1.7383&2.5911&22.966&4.978\\ \hline 0.235&4.257&3.257&1.9596&2.9148&21.986&5.165\\ \hline 0.255&3.928&2.928&2.2085&3.2763&20.969&5.339\\ \hline 0.276&3.624&2.624&2.4880&3.6788&19.915&5.496\\ \hline 0.299&3.344&2.344&2.8017&4.1255&18.824&5.630\\ \hline 0.324&3.085&2.085&3.1532&4.6192&17.695&5.736\\ \hline 0.351&2.847&1.847&3.5465&5.1620&16.530&5.807\\ \hline 0.381&2.627&1.627&3.9854&5.7549&15.331&5.837\\ \hline 0.413&2.423&1.423&4.4740&6.3975&14.099&5.818\\ \hline 0.447&2.236&1.236&5.0163&7.0873&12.837&5.741\\ \hline 0.485&2.063&1.063&5.6159&7.8196&11.550&5.598\\ \hline 0.525&1.904&0.904&6.2758&8.5867&10.242&5.380\\ \hline 0.569&1.756&0.756&6.9985&9.3782&8.921&5.079\\ \hline 0.617&1.621&0.621&7.7854&10.1809&7.594&4.686\\ \hline 0.669&1.495&0.495&8.6369&10.9798&6.269&4.192\\ \hline 0.725&1.380&0.380&9.5520&11.7587&4.954&3.591\\ \hline 0.786&1.273&0.273&10.5285&12.5023&3.661&2.875\\ \hline 0.851&1.175&0.175&11.5628&13.1967&2.396&2.040\\ \hline 0.923&1.084&0.084&12.6507&13.8316&1.168&1.078\\ \hline 1.000&1.000&0.000&13.7872&14.3999&0.000&0.000\\ \hline \end{array}}$$

Last edited: Jan 4, 2015
4. Jan 10, 2015

### TEFLing

I want to try to explain in my own words

Imagine you have some cosmological measurement device, of some fixed size L

With two lights on the two ends, such that the beacons are just now becoming visible

At z=1.6, you could fit 2*pi*5.837Glyr/L of those devices in a ring around our present position

At z=4, you could only fit 2*pi*4.87Glyr/L of those devices in a ring around our present position

Thus they must look larger on the sky, each occupying more degrees of arc, such that fewer add up to 360

5. Jan 10, 2015

### marcus

Agreed! and at z = 1090 (the ancient glow from early universe hot gas) they would look enormous by comparison, and you could only fit far fewer : ^)
The "4.87 billion LY" in your z=4 expression would have to be replaced by something like 42 million LY. That is, more than 100 times smaller.
So you could only fit 100 times fewer of the standard rods in a ring around us, and they would look 100 times bigger in the sky. Compared with the z=4 case.
I'm not sure my adding that redundant comment makes anything clearer. Your example already says it right enough.

6. Jan 10, 2015

### marcus

Maybe one could add that sound waves in the ancient hot gas are frozen in, fossilized, in the form of temperature fluctuations of the CMB. Since astronomers know a fair amount about the conditions in that gas at the epoch when it became transparent they can estimate the speed of sound in that glowing gas and what the wavelengths should be. So the wavelengths of pressure-wave ripples can serve as the "L" in your example---they can serve as a standard measure rod, or linear scale.

And they look enormous in the sky. Because the radius of your imagined ring is only about 42 million LY. So the universe has a kind of "fun house" optics where things look what seems at first to be the wrong size. But it actually helps astronomers figure things out and keep track of the expansion history.

7. Jan 11, 2015

### krater

All very informative and illuminating posts. Many thanks to both TEFLing and marcus for some excellent insights!

8. Jan 12, 2015

### TEFLing

If the major temperature fluctuations in the CMB, seen on the sky, are about half of a degree across...

Since one radian is equal to 50-60 degrees

So multiplying the proper distance times the (sine of the tiny) angle...

Gives ~0.4 Mlyr

So as time passes, and more and more CMB photons arrive, from gradually but inexorably and progressively farther away...

Shouldn't the CMB fluctuations seem to evolve in time, on time scales of hundreds of thousands of years?

If so, then from the first Homo Erectus living ~2 Myr ago until today, humans (archaic and modern) have witnessed (if mostly unknowingly) something like 5 CMB fluctuations, on each patch of sky, evolve into and then out of view

(?)

9. Jan 12, 2015

### TEFLing

For clarity I'm trying to understand the actual physical metric size, of the temperature fluctuations in the early pan cosmic medium, which gave rise to the CMB fluctuations we see on the sky today...

At z~1e3 ages and eons and long long ago ...

Back at the time the CMB photons first started freely streaming towards our region of space...

0.01 * 42 Mlyr ~= 0.4 Mlyr

10. Jan 12, 2015

### Staff: Mentor

On a scale of hundreds of millions of years. Those 0.4 Mly then got blown up to 400 Mly now.

11. Jan 12, 2015

### TEFLing

Since the initial 0.4 Mlyr of space, between the front and back of a pressure / temperature wave fluctuation (as seen from earth)...

From which photons were emitted then...

Has since grown into 1000x the distance?

12. Jan 12, 2015

Right.