Object moving out of the Observable Universe due to Expansion

[MLaw]

With the LDCM, cosmological constant, model I understand that the scale factor of the Universe grows more rapidly than the Horizon. I believe the correct horizon I need to be considering is the Hubble Horizon and the point when objects recessional velocity hits the speed of light they disappear from our view. From this I'd expect to be able to calculate the time it would take for an object at some distance from us to move to this point.
Is there a neat equation for this or is there an issue with my understanding of what's happening?

If it makes sense but there's no equation that can be given, could somebody give me some steps on how to begin?

I also came across this calculator, which gives a recessional velocity of an object at redshift 1.65 to be greater than the speed of light. Have I interpreted the definitions wrong or is that the limit of our observible universe?
http://www.einsteins-theory-of-relativity-4engineers.com/cosmocalc_2010.htm

I am a third year physics student so I'm pretty comfortable with mathematics and if you think something is a bit beyond my level I'd still like to take a look.

Thank you

 

[phinds]

With the LDCM, cosmological constant, model I understand that the scale factor of the Universe grows more rapidly than the Horizon. I believe the correct horizon I need to be considering is the Hubble Horizon and the point when objects recessional velocity hits the speed of light they disappear from our view.

Objects currently near the edge of our Observable Universe are currently receding from us at about 3c. The light that we are seeing from them in our "now" were omitted long ago and the light that they are emitting in their "now" will reach us in about 48 Billiion years (but it will be very red shifted and hard to detect).

 

[MLaw]

Do you know specifically how we come to the 3c? I'm looking for some sort of formulas or graphical representation of what's going on so I can really grasp what's going on.

 

[Bandersnatch]

Have I interpreted the definitions wrong

Looks that way. The Hubble radius is not a cosmological horizon. A horizon means the boundary from beyond no signal can ever reach the observer.
If you consider a universe like ours, in which the Hubble constant goes down in time, asymptotically approaching some value, then it follows that the Hubble radius is increasing. An object currently at the $H_R$ recedes at c and emits light, also traveling at c, towards the observer. This light would never reach the observer only if the Hubble radius were constant, in which case the light would 'hover' at that distance, never gaining any ground as the expansion would carry it away at a rate of 1ly/y - just as it moved ahead at the same rate.
If the Hubble radius grows, then light emitted at $H_R$ or somewhat beyond it will soon find itself in a region that recedes at less than c, and it will be able to make its approach.
The asymptotic growth means that in the far future, when the matter content of the universe will have diluted sufficiently for the expansion to be approximated by a de Sitter universe (i.e. as if containing only dark energy), then the $H_R$ will be functionally equivalent to a horizon.

I'm looking for some sort of formulas or graphical representation of what's going on

The calculator you linked to has an option of graphing the light cones and other data, such as evolution of the Hubble radius. Try playing with the display and column selection options.
It was made by our forum member Jorrie - there's a sticky thread about it in the cosmology section, and he wrote a series of tutorial insights, including on how things are calculated:
https://www.physicsforums.com/insights/?s=lightcone7

This paper:
https://arxiv.org/abs/astro-ph/0310808
includes the following light cone graphs:

showing clearly that the Hubble radius is not the event horizon.

The version below includes recessional velocities: (sadly I don't know whom to credit for it)

The following is an animated version made by yukterez:
http://yukterez.net/lcdm/lcdm-flrw-animation.gif

This PF inshight article includes some helpful diagrams and relevant equations:
https://www.physicsforums.com/insights/inflationary-misconceptions-basics-cosmological-horizons/

Objects currently near the edge of our Observable Universe are currently receding from us at about 3c. The light that we are seeing from them in our "now" were omitted long ago and the light that they are emitting in their "now" will reach us in about 48 Billiion years (but it will be very red shifted and hard to detect).

The last part of it is not true, though. The objects at the proper distance equal to the particle horizon (i.e. the parts of the universe that emitted the CMBR are now) are already beyond the event horizon. The light they emit now will never reach us. This is clearly apparent on the graphs above - try tracing the light-like worldlines from emitters at the intersection of the 'now' and the 'particle horizon' lines.

 

[phinds]

The last part of it is not true, though. The objects at the proper distance equal to the particle horizon (i.e. the parts of the universe that emitted the CMBR are now) are already beyond the event horizon. The light they emit now will never reach us. This is clearly apparent on the graphs above - try tracing the light-like worldlines from emitters at the intersection of the 'now' and the 'particle horizon' lines.

OK. Also, I just realized that even if it DID reach us it certainly wouldn't be in 48 billion years but rather that plus more due to expansion.

 

[Chronos]

Even bodies at the very 'edge' of the observable universe will never actually 'jump the fence' and someday just abruptly wink out of causal contact with our instruments. They will just boringly continue to redshift into obscurity, not unlike Alice watching Bob fall into a black hole. It matters not [to Alice] when Bob is, or is not, eaten: she will never see it happen.

 

[Chalnoth]

Objects currently near the edge of our Observable Universe are currently receding from us at about 3c. The light that we are seeing from them in our "now" were omitted long ago and the light that they are emitting in their "now" will reach us in about 48 Billiion years (but it will be very red shifted and hard to detect).

Really? Intuitively that doesn't seem right to me. The expansion rate won't slow by very much in the future, so I would tend to expect that objects that currently have recession velocities greater than about 1.3c or so are beyond our horizon. Where did you get the 48 billion years from?

 

[phinds]

Really? Intuitively that doesn't seem right to me. The expansion rate won't slow by very much in the future, so I would tend to expect that objects that currently have recession velocities greater than about 1.3c or so are beyond our horizon. Where did you get the 48 billion years from?

See post #5

 

[Chalnoth]

See post #5

Ahh, my bad.

For a bit of back-of-the-envelope reasoning on why I expect objects with recession velocities greater than about 1.3c to be beyond the horizon:

In the far future, if our universe has a cosmological constant, then the horizon will eventually settle to the location where the recession velocity = c. To see this, consider that if there is only a cosmological constant, then a photon emitted from a location with recession velocity = c traveling in our direction will, from our perspective, never move from that spot. This is because the rate of expansion doesn't change in such a universe.

Right now, the Hubble expansion factor is roughly 70km/s/Mpc, and the dark energy density fraction is roughly 0.7 (not getting exact numbers because this is just a very rough estimate). Thus, in the far future, the rate of expansion will decrease from the current 70km/s/Mpc to $\sqrt{0.7} \times 70$km/s/Mpc $\approx 60$km/s/Mpc. Thus in the far future the horizon will be located about 20% further than the current location where the recession velocity is c. I'd expect the current horizon would be located at close to this position, which is where the recession velocity is approximately 1.2c today (though I'd have to think more carefully to determine whether it would be closer or further away than the eventual horizon: I suspect closer).

 

[Bandersnatch]

I'd expect the current horizon would be located at close to this position, which is where the recession velocity is approximately 1.2c today (though I'd have to think more carefully to determine whether it would be closer or further away than the eventual horizon: I suspect closer).

Or, you could just look at the graphs from the Davis&Lineweaver paper. For more precision, Jorrie's calc outputs 16.5 Gly.
So, indeed closer.

 

[MLaw]

Thanks for all your responses, I'll spend a little while trying to get my head around all of this. I wanted the question to stay general at first so I could get a better understanding. But the specific thing I'm trying to solve is;
Objects currently at a distance with redshift 2 are being measured for some experiment to determine the nature of dark energy, I want to show that with a cosmological constant that at some point future observers would be unable to perform the same experiments because the objects required to do this have disappeared out of view. I've seen numerous lecture notes including my unis own and a couple documentaries saying that if we have a barotropic parameter < -1/3 then the scale factor increases more rapidly than the horizon so objects should disappear out of view. Chronos's post says that isn't the case and things just redshift into obscurity. I passed this idea past my lecturer and he seemed to like it, can anyone confirm that my idea makes physical sense?

 

[Bandersnatch]

I've seen numerous lecture notes including my unis own and a couple documentaries saying that if we have a barotropic parameter < -1/3 then the scale factor increases more rapidly than the horizon so objects should disappear out of view.

There's a difference between objects leaving the event horizon, and their light doing the same.
Take a closer look at the comoving distance vs conformal time graph in the post #4 above. While it's true that galaxies (each vertical comoving distance line) are constantly leaving the event horizon, the light they emit just before doing so reaches the observer only at infinite future. This means that in principle, barring detectability issues, you never lose sight of any galaxy that has ever been inside the event horizon.

 

[MLaw]

Ah, that does solve the issue in my mind.
Once again, I massively appreciate all your input.