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Cosmological Horizons

Cosmological Horizons vs. Standard Theory

The basic idea of a cosmological horizon seems to be straightforward: if the universe is expanding, then the further away an object is, the faster it'll be receding from us. Beyond a certain distance, objects might be expected to have a recession speed greater than lightspeed, which would lead us to expect that we wouldn't be able to see them. The critical distance from an observer would therefore seem to mark out an observational horizon

However, when we get into this subject more deeply, things don't turn out to be quite so straightforward.

Argument #1: "Cosmological horizons must exist "

In an evenly-expanding universe, the mutual recession between two chosen points on the surface depends on how far they are apart. For a given distance, the cosmological recession velocity will be half lightspeed ... at double that distance, it'll be lightspeed .... at double that distance, the nominal recession will be twice lightspeed.

In a spatially-closed expanding universe, it'd seem that we can always find objects at the critical distance, even if our universe is very small, and so that we have to "loop the loop" around the universe to get there.

Argument #2: "Cosmological horizons cannot exist"

In a spatially-closed universe with a reasonably-dense, reasonably-even distribution of matter, any two adjacent nearby stars ought to be capable of seeing each other and being in contact with each other. But if cosmological horizons exist, then any point in space could be said to be on a cosmological horizon with respect to a suitably-chosen viewing position. For any given pair of stars, there'll be a viewing position somewhere whose  nominal cosmological horizon passes between the two stars, with one star is reckoned to be inside the horizon, and the other one outside.
And yet ... signals still manage to pass between them, in both directions
Mainstream response #1: "Special relativity's velocity-shift relationships don't apply to cosmological shifts"
A distant object on our side of a cosmological horizon is quite capable of accelerating to a location on the other side, which sounds as if it might cause problems for special relativity's idea that an object can't increase in velocity past someone else's speed of light. Cosmologists counter that special relativity's arguments (and its shift relationships) don't apply when cosmological curvature is in play, because cosmology isn't a flat spacetime problem. But it spoils general relativity's efficiency to have two different effects that look the same, and associate similar effects with recession velocities, but which apply those effect according to two different laws. It would seem that a distant receding redshifted galaxy can have two different velocities, a Hubble recession velocity and a local velocity compared to its immediate environment, with the two types of recession ("velocity" and "Hubble velocity") each generating different Doppler characteristics. This doesn't seem to be a very satisfactory situation.
Mainstream response #2: "Cosmological horizons don't exist"
We're sometimes told that the recession velocity-shift relationship for cosmological effects is freq'/freq = c / (c + vHUBBLE), which doesn't produce a zero observed frequency for any finite value of vHUBBLE.  This would allow signals to reach us even from objects that are nominally "Hubble-receding" at more than the speed of light. But if we accept this answer, it'd suggest that mainstream work that does involve cosmological horizons was wrong. There'd then seem to be some disagreement between experts as to the correct way to tackle these problems.

Argument #3: "Cosmological horizons must exist"

If the age of the universe (measured using "insider-time") is finite, then as we look further and further away, and see events that took place progressively longer ago, we should eventually reach a limit that corresponds to the initial event(s) at the universe's origin. We should be able to plot a radius around us at which the events that we'd be seeing would correspond to the Big Bang. We shouldn't be able to see any further back or any further away than this, because there shouldn't be any events inside our universe before this, and there also shouldn't be any space before this for events to take place in. If observerspace coordinates seem to stop dead at or before this limit, then for this to happen in a well-behaved way, we seem to need the observational limit to represent a coordinates where timeflow appears frozen ... that is, we need our spatial limit to observation to behave like an event horizon.

A variation on this argument describes the hypothetical initial origin-point of the universe as a singularity, which then triggers the cosmic censorship hypothesis. According to the c.c.h., singularities (which represent the limit at which physics appears to break down) aren't allowed to be "visible". If they exist, they must always be hidden from us by an intervening horizon ("no naked singularities").

Considering the problem in terms of the censorship hypothesis, and considering the viewable universe as being topologically equivalent to a black hole turned inside-out, leads us to the idea that perhaps effects due to the cosmological curvature of spacetime might be expressed (with a topological twist) using language and ideas that were originally developed for the gravitational curvature of spacetime.

Gravitation-equivalence

As the universe expands, it's mass-density reduces. This means that as we look at more and more distant objects, we're effectively looking back in time to a period in which the background gravitational field strength was higher. In literal observerspace descriptions, we should see what appears to be a gravitational field strength that increases with distance, and we should see the more distant stars seeming more redshifted,  apparently as a function of this gravitational field, and/or as a function of their recession. We can also argue that (in the observerspace description) the apparent existence of the field should be associated with a freefall motion of stars embedded within it, so that we should see distant galaxies receding from us, and receding faster the further away they are. This is simply an alternative geometrical description of the same expanding-universe model that we started out with. It's the same physics looked at from a different perspective.

At a certain distance, the field strength difference between "here-and-now" and "there-and-then" will be expected to have an escape velocity equal to or  greater than lightspeed, and at that point we should expect the redshift to become total. Within the observerspace description, this horizon then looks like a gravitational horizon surrounding a high-gravity object.

If we take our descriptions of the redshift distribution due to a black hole, and due to spatially-closed cosmological model, we can combine them to produce a single descriptions. This is a powerful method, as if means that cosmological effects have to be bound by gravitational theory, and vice versa

While it is very satisfying to be able to predict the same effects in different ways, this trick doesn't seem to work with our current implementation of general relativity. While we might like to model the same horizon using cosmological arguments and gravitational arguments, current GR can't manage it. Cosmological horizons are surrounded by radiation that the observer can;t see directly, but which appears when they accelerate through a region (see Unruh radiation) allow indirect radiation. Cosmological horizons leak.

GR1915's gravitational horizons are totally sealed, and completely cut a region off from the outside world. If we want to say that current general relativity is the correct classical description, then gravitational horizons would have to operate according to different rules to cosmological horizons. If we wanted to unify both descriptions, then since cosmological horizons have to leak, gravitational horizons would need to leak, too, and according to 1960's relativity theory, this was supposed to be provably impossible.

The situation shifted slightly in the 1970's, with the idea of Hawking radiation ...

According to modern quantum mechanics, spacetime around an event-horizon-bounded region should radiate, and according to 21st-Century quantum mechanics, the patterns embedded in that radiation are now supposed to belong to information that would be reckoned to have previously been somewhere behind the horizon.
As a result of Stephen Hawking's 1970's arguments, we now believe that gravitational horizons should be "leaky" after all, and we might think that this solves the incompatibility problem mentioned earlier. Hawking radiation was originally considered to be a strictly "quantum" effect, but as we studied the relationships in more detail, we found that the same basic statistical patterns also happened in other situations due to entirely classical mechanisms, and one of the situations involving classical Hawking radiation was radiation through a cosmological horizon.

The idea of cosmological Hawking radiation seems to dissolve the apparent contradiction between our earlier arguments that cosmological horizons have to exist, but that information has to be capable of passing across them. If we look at the old classical indirect-radiation mechanism, and compare them to the behaviour of cosmological horizons, we find that a cosmological horizon actually counts as a category of acoustic horizon ... it's a mathematical surface that theoretically describes observational limits, but the surface fluctuates and fizzes in response to local events, some of which are behind the horizon surface. An acoustic horizon is descriptive rather than prescriptive -- it doesn't dictate the physics, or the events that are allowed to happen, it instead jumps about mathematically in response to them.
The "statistical mechanics" of acoustic horizons involves fluctuation and radiation, just like the "quantum mechanics" description of Hawking radiation, and acoustic horizon radiation is now considered a legitimate instance of Hawking radiation.

If we now believe that cosmological horizons and gravitational horizons both radiate, does this mean that we can now finally unite the two descriptions and solve our problem?

Not quite, because while the cosmological radiation effects are "mundane" and operate according to straightforward classical physics, current GR has no way of replicating the effect classically. To get a GR black hole to radiate we have to invoke Hawking radiation effects separately, and put the effect in "by hand" using quantum mechanics. In the "cosmological horizon" case, QM is describing the statistical results of classical behaviour, but in the GR+QM black hole case, there's nothing within GR for the QM statistics to relate to, we simply say that since the QM result has to be correct, the radiation must be due to some of those spooky counterintuitive effects that QM descriptions sometimes produce.
The difference between these two descriptions seems to be due to the choice of underlying metric, the classical indirect-radiation effect happens in the context of acoustic metrics, while general relativity instead reduces over small regions to the Minkowski metric of special relativity. We can imagine a different general theory that reduces to an acoustic metric (and which would therefore seem to allow indirect radiation and solve the black hole information paradox) but since the relationships of the metrics are different, this would seem to mean stepping away from special relativity to a different sort of relativistic model, and if we've been trained to see relativity theory through  they eyes of current physics textbooks, we'll have no idea what such a "non-SR" theory of relativity would look like.
Also, if we were to combine the cosmological and gravitational descriptions, we'd really need the resulting unified description to apply the same velocity-shift relationship to both situations, and if the cosmological case can;t use the shift relationships of SR, that'd mean that gravitational theory couldn't use them either.
Since we'd need a set of relationships that could reproduce gravitational horizons, but which wouldn;t be special relativity, parts of our gravitational theory and parts of our cosmological theory would be wrong.

Conclusions

Although Einstein's general theory was eventually applied to the case of a spatially-closed, expanding universe, it was not originally devised with this sort of universe in mind. The theory was not designed to make use of relativistic and geometrical arguments that only make sense in a spatially closed universe, such as the gravitation-equivalence idea mentioned above.

The "cosmological horizon" problem is an interesting one, in that it seems to break current theory, and also seems to suggest how a replacement theory might be constructed. To solve the "horizon problem", we seem to be required to switch from special relativity's Minkowski metric to an acoustic metric, which in turn would seem to require us to implement the idea of local lightspeed constancy, and the principle of relativity in inertial physics, with gravitomagnetic arguments rather than with special relativity's flat spacetime. If the mathematical horizon at a cosmological threshold fluctuates and jumps about in response to medium-scale and small-scale events there, then "cosmological horizon physics" must allow lightspeeds to be affected in a fundamental way by the motion of bodies, even when those bodies wouldn't normally be said to have significant gravitational fields. And since any point in the universe could be considered as being at a cosmological horizon for someone, these cosmological arguments for the variability of lightspeeds should also apply to physics here on Earth.

"Academic" questions that at first sight would seem to be remote from everyday experience and testing ("What is our view of ancient events at the other side of the universe, and of cosmological-scale features?") can end up telling us important things about our own physics ("If I throw this baseball, does it affect the speed of light in the adjacent region?").

References
• Michael A. Seeds, Foundations of Astronomy, Third Edition (1992)
Box 6.1 The Hubble Law " ... the more distant a galaxy, the faster it recedes from us.  A galaxy's radial velocity in kilometers per second equals the Hubble constant times its distance in megaparsecs.  ...  Vr = HD  ... The best measurements of distance and velocity suggest that H is approximately equal to 70 km/sec/Mpc."
• Eric Baird, Relativity in Curved Spacetime (2007) section 12 "What's wrong with general relativity?", especially 12.11 "Do cosmological horizons count as 'acoustic'?" and 12.13 "Grand unification?"
• For the reasons given above, trying to find a definitive statement on the status of cosmological horizons under standard theory can be difficult. For a critical overview of some apparently conflicting views, see:
• Tamara M. Davis and Charles H. Lineweaver, "Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe"  Publications of the Astronomical Society of Australia (PASA) 21 97-109 (2003) astro-ph/0310808"We use standard general relativity to illustrate and clarify several common misconceptions about the expansion of the Universe. To show the abundance of these misconceptions we cite numerous misleading, or easily misinterpreted, statements in the literature. In the context of the new standard Lambda-CDM cosmology we point out confusions regarding the particle horizon, the event horizon, the ``observable universe'' and the Hubble sphere (distance at which recession velocity = c). ...
• Early work by Zel'dovich on particle-production across a field gradient was cited in "MTW"'s Gravitation chapter 30, p.816 (1973) as suggesting a possible solution to problems involving causality and cosmological horizons. The subject of quantum mechanics and indirect-radiation effects seems to have been tentatively considered for the case of cosmological horizons shortly before Hawking's work applied similar arguments to gravitationally horizon-bounded bodies.
• For references on Hawking radiation (quantum and classical) see the references section at hawking_radiation.html .
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