I The Boundary of the Observable Universe

[PeterDonis]

if something is outside our observable universe, the photons it emitted even at the beginning of the universe would not have had time to reach us

Only if it is a comoving object. That is @Ibix's point. If the object is not comoving, this statement you are making is no longer always true. It is possible for a non-comoving object to be outside our particle horizon now, but to have a portion of its worldline in our past light cone now.

 

[Jaime Rudas]

Only if it is a comoving object. That is @Ibix's point. If the object is not comoving, this statement you are making is no longer always true. It is possible for a non-comoving object to be outside our particle horizon now, but to have a portion of its worldline in our past light cone now.

Could you provide a concrete numerical example of this?

 

[PeterDonis]

Could you provide a concrete numerical example of this?

Consider an object that, at the "now" surface of the bottom diagram of Fig. 1 in Davis & Lineweaver 2003 (the one that uses conformal time), is at a comoving distance of 50 Gly. That puts it outside the particle horizon.

Now draw a worldline from that point that goes down and inward, decreasing in comoving distance, so that, at the initial moment of time (the bottom line of the diagram), it is at a comoving distance of 20 Gly. That is a timelike worldline (because its slope is closer to vertical than 45 degrees, and on a conformal diagram, that is timelike), and it has a substantial segment that is within our past light cone "now", so we will be able to see that portion of its history.

 

[Bandersnatch]

And that works to at most twice the distance of the particle horizon, as beyond that distance not even an object moving away at arbitrarily close to light speed could ever have been inside our past light cone.
Since I've never heard of this distance be called anything, I call dibs on naming rights. From now on it will be known as the particlest horizon :P

 

[Orodruin]

From now on it will be known as the particlest horizon

If there is a particle horizon and a particlest horizon there must be a particler horizon …

 

[Bandersnatch]

If there is a particle horizon and a particlest horizon there must be a particler horizon …

That's anything in-between, duh. It's not very particular.

 

[Ibix]

I don't think I understand the figure you propose, but the highlighted part seems contradictory to me because any point I choose within our past lightcone will necessarily be within our observable universe.

The above is what I had in mind. The diagram is the third part of Davis & Lineweaver figure 1, to which I have added a red line. That line represents a non-comoving timelike object that starts inside our current past lightcone but has not yet crossed the particle horizon. It is literally observable, but is not inside the observable universe as defined by the intersection of the particle horizon and the current cosmological time surface. Additionally, it's clearly possible to draw a timelike path that starts at the same point as my red line and never crosses the particle horizon, at least given this model universe's parameters.

The velocity drawn above is fairly extreme as peculiar velocities go. Our own peculiar velocity is about 600kps, or about 0.002c. Assuming that peculiar velocities are such that correctly calculated lines like my red one aren't too curved (or, at least, not too curved in the wrong direction), such objects whose origin points are just now inside our past light cone could now be around $0.002×14\mathrm{bn}\approx 30\mathrm{m}$ light years outside our current particle horizon. So that would suggest that objects forming in the early universe could be observable for several million years before entering the observable universe by the "current particle horizon" definition.

The integrator code I posted in the thread about circumnavigating closed FLRW universes could be modified to do the correct form of my simple linear calculation and draw the red line accurately.

 

[Jaime Rudas]

Consider an object that, at the "now" surface of the bottom diagram of Fig. 1 in Davis & Lineweaver 2003 (the one that uses conformal time), is at a comoving distance of 50 Gly. That puts it outside the particle horizon.

Now draw a worldline from that point that goes down and inward, decreasing in comoving distance, so that, at the initial moment of time (the bottom line of the diagram), it is at a comoving distance of 20 Gly. That is a timelike worldline (because its slope is closer to vertical than 45 degrees, and on a conformal diagram, that is timelike), and it has a substantial segment that is within our past light cone "now", so we will be able to see that portion of its history.

The above is what I had in mind. The diagram is the third part of Davis & Lineweaver figure 1, to which I have added a red line. That line represents a non-comoving timelike object that starts inside our current past lightcone but has not yet crossed the particle horizon. It is literally observable, but is not inside the observable universe as defined by the intersection of the particle horizon and the current cosmological time surface. Additionally, it's clearly possible to draw a timelike path that starts at the same point as my red line and never crosses the particle horizon, at least given this model universe's parameters.

I finally managed to understand how this is possible. Indeed, it turns out that the object we observe is now outside our observable universe, but what we actually observe is what the object was like when it was located at a coordinate that is now within our observable universe.

 

[Ibix]

I finally managed to understand how this is possible. Indeed, it turns out that the object we observe is now outside our observable universe, but what we actually observe is what the object was like when it was located at a coordinate that is now within our observable universe.

Located at a spatial coordinate that is now within our observable universe, sure. But I would say this is essentially using the definition of observable universe discussed in section 3.4 of Davis and Lineweaver, projecting the particle horizon now back in time along comoving worldlines. I think you pointed out this discussion in a now-deleted post.

 

[PeterDonis]

I would say this is essentially using the definition of observable universe discussed in section 3.4 of Davis and Lineweaver, projecting the particle horizon now back in time along comoving worldlines.

But the point of that definition is to be able to make use of the ordinary implication of the word "observable"--that if an object is outside our observable universe, we cannot have observed any light from it at all, from any portion of its history. But that implication is only valid for an object whose worldline is comoving.

If, on the other hand, we use the term "past light cone", then we have a straightforward rule: any event in our past light cone is observable, so any object that has a portion of its worldline in our past light cone is observable for at least a portion of its history. But for whatever reason, cosmologists don't seem to want to approach things that way.

 

[Ibix]

But the point of that definition is to be able to make use of the ordinary implication of the word "observable"--that if an object is outside our observable universe, we cannot have observed any light from it at all, from any portion of its history.

And Davis and Lineweaver don't like that definition either (although they cite several sources using it) preferring to define the particle horizon as the future lightcone of our origin event on the singularity, as in the chart I copied in #37 (again citing sources using that definition).

But that implication is only valid for an object whose worldline is comoving.

Quite. And this is the argument I was trying to make about Tong's "nothing outside the particle horizon can influence us today" (see #20). I think it's possible to read this the way @Jaime Rudas does in #25, but in that case it's only correct in the idealised model. Reading it as "our past light cone" is generally true, and it's not

But for whatever reason, cosmologists don't seem to want to approach things that way.

To be fair, it would not be the only bit of physics jargon that developed organically and doesn't make complete sense when examined closely. And it's also worth noting that typical peculiar velocities are sufficiently low that statements like "the furthest object in the observable universe is currently 46Gly away" remain true whatever definition you adopt.

 

[Jaime Rudas]

Located at a spatial coordinate that is now within our observable universe, sure. But I would say this is essentially using the definition of observable universe discussed in section 3.4 of Davis and Lineweaver, projecting the particle horizon now back in time along comoving worldlines. I think you pointed out this discussion in a now-deleted post.

No, no. What I am saying is that the definition of the current observable universe used in cosmology does not correspond to "the set of objects that are the source of the photons that we can theoretically observe today" but rather corresponds to "the set of comoving coordinates from which, in theory, the photons that we can observe today could have been emitted" or, what is the same, "the set of comoving coordinates from which, in theory, the photons that had enough time to reach our position could have departed."

What is discussed in section 3.4 of Davis and Lineweaver is something else: that it is not correct to represent the evolution in time of the particle horizon as the worldline of the coordinate of the current particle horizon, but rather it is necessary to represent it as the distance at which, at each moment, the particle horizon is located. That is, the distance to the particle horizon at time $t$ is not equal to $R(t) \chi_{ph}(t_0)$ but is equal to $R(t) \chi_{ph}(t)=cR(t) \int_0^t \frac {dt'}{R(t')}$

 

[Bandersnatch]

I find the whole preoccupation with what does the emitter do with its life after emission to invite more confusion than it purports to solve.
How far could a signal have travelled in the expanding universe? - the particle horizon. From how far a signal can reach us? - the particle horizon. What is the observable universe? - the particle horizon. Is it essentially the same as the largest extent of a light cone? - the particle... I mean, yes, yes it is.
What do I care about the rest of the worldline here?
If I get a letter from Australia wherein I'm informed the sender is going to get on a boat and row real hard to reach the land where devils dwell, I still got a letter from Australia. That the bendy worldline of the sender at some future (post-emission) point intersects Tasmania doesn't change the fact the letter bears a stamp of the Sydney Post Office. It's got a picture of the chief bogan strangling a kangaroo and all that shtick.

 

[PeterDonis]

To be fair, it would not be the only bit of physics jargon that developed organically and doesn't make complete sense when examined closely.

Yes, I definitely agree with this.

And it's also worth noting that typical peculiar velocities are sufficiently low that statements like "the furthest object in the observable universe is currently 46Gly away" remain true whatever definition you adopt.

Remain approximately true. You calculated an approximate error of around 30 Mly, which is indeed small, less than a tenth of a percent of the particle horizon comoving distance, but not zero, and not outside the range of differences that we can observe. For example, 30 million years after the initial surface in the diagram in Davis & Lineweaver is well after the CMBR surface of last scattering and therefore in a region where visible light can propagate freely, so the only bar to us seeing portions of the history of objects within our past light cone in that range is the sensitivity of our instruments, which keeps improving.

 

[Jaime Rudas]

But the point of that definition is to be able to make use of the ordinary implication of the word "observable"--that if an object is outside our observable universe, we cannot have observed any light from it at all, from any portion of its history. But that implication is only valid for an object whose worldline is comoving.

If, on the other hand, we use the term "past light cone", then we have a straightforward rule: any event in our past light cone is observable, so any object that has a portion of its worldline in our past light cone is observable for at least a portion of its history. But for whatever reason, cosmologists don't seem to want to approach things that way.

Perhaps it should be noted that there is no way to determine the peculiar velocity of distant galaxies, so it is reasonable to assume that it is zero on average.