# I Can Hyperbolic Space be affecting our view of the universe?

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1. Sep 29, 2016

### particlezoo

Usually when gravitational lensing is discussed, the examples are those of matter bending spacetime into a positive curvature.

https://commons.wikimedia.org/wiki/File:Gravitational_lens-full.jpg

In these cases, distortion of light is clearly evident as images of galaxies from behind these "gravitational lenses" are warped drastically. The below is one clear example of this distortion:

https://commons.wikimedia.org/wiki/File:A_Horseshoe_Einstein_Ring_from_Hubble.JPG

There appears to be far less mention of the idea of hyperbolic space curvature, which is has negative curvature in the Gaussian sense. To illustrate what this looks like, here is an depiction of a Hyperbolic 3-manifold:

https://en.wikipedia.org/wiki/Hyper...erbolic_orthogonal_dodecahedral_honeycomb.png

Based on what I have read on hyperbolic space, it doesn't seem to me that the Hyperbolic space would cause significant (obvious) distortions like we see in examples of gravitational lensing. If I understand correctly, hyperbolic space would not causes images of celestial objects to flip upside down, nor would they turn them into highly-distorted rings. However, it does seem that it would cause objects to appear farther than the really are.

To get an appreciation of hyperbolic space, the folks from Numberphile posted a few videos on the subject of hyperbolic space:

Playing Sports in Hyperbolic Space - Numberphile

More Hyperbolic Sports - Numberphile

Now for the big question:

How significant could this hyperbolic curvature be without it being obvious that it exists? To take this idea to the limit, could we actually be in an universe where galaxies, or even stars, are orders of magnitude closer than we think, and that the only reason why it does not appear that way is optical trickery due to negatively curved space?

Can Hyperbolic Space be affecting our view of the universe?

Sincerely,

Kevin M.

2. Sep 29, 2016

### Twigg

I believe the answer to your question is a definite no, unless I am misunderstanding. Flat spacetime is described by hyperbolic space, which has zero intrinsic scalar curvature in the sense of a Lorentzian manifold (a Riemannian manifold having a signed metric). A 2-dimensional hyperboloid has negative Gaussian curvature, which measures instrinsic sense in the sense of a Riemannian manifold (a manifold having a positive-definite metric), and zero instrinsic curvature as a Lorentzian manifold. These are two incompatible measures of intrinsic curvature, if I'm not mistaken.

The issue is that hyperbolic space is the simplest way in which we can perceive the universe. Hyperbolic spacetime isn't weird, but a (4-dimensional) Euclidean spacetime would be very weird. For instance, in a Euclidean spacetime Maxwell's equations in free space would become a 4D analog of Laplace's equation instead of wave equations, so light would propagate in a very different unintuitive way. Technically, you could say that hyperbolic space affects how we perceive the universe, in that it gives us the most familiar physics we have.

3. Sep 30, 2016

### Staff: Mentor

The curvature caused by an isolated massive object is only positive in certain directions. It's negative in other directions. The directions in which it is positive happen to be the ones that come into play in gravitational lensing.

The longer answer has several parts. First, it's important to distinguish spacetime curvature from space curvature. You use both terms in your post, but it seems like you are really only thinking of space curvature. But to really understand gravitational lensing, as well as other possible optical effects, you need to think of spacetime curvature.

Spacetime curvature is tidal gravity; you observe it by observing how geodesics (worldlines of freely falling particles or light rays) converge or diverge. Converging geodesics, as in the case of gravitational lensing, indicate positive spacetime curvature. Diverging geodesics indicate negative spacetime curvature. So to see how negative spacetime curvature works, we have to find an example of diverging geodesics.

One example of such is the following: suppose we have two objects that are freely falling radially near a large mass like a planet or star, and are separated radially but in no other direction. The worldlines of these objects will diverge. Similarly, two light rays moving radially near a large mass will diverge. But this isn't something that will be visible optically from a distance, because the motion is radial, not tangential; there won't be any "inverse lensing" effects because of this.

The other possibility would be effects due to the geometry of spacetime on the scale of the universe as a whole. On that scale, however, the only significant spacetime curvature is that due to the expansion of the universe, or more precisely the way in which the expansion decelerates (positive curvature) or accelerates (negative curvature) over time. This does affect how we assign distances to objects in the universe, but not in the way you describe; measurements of the universe's expansion rate and how it changes with time are key inputs to how we determine the distances to objects in the universe, because they tell us how those distances are related to the apparent brightness of objects.

Ned Wright's cosmology tutorial has a good discussion of the basics of how all this works:

http://www.astro.ucla.edu/~wright/cosmo_01.htm

4. Sep 30, 2016

### particlezoo

Yes, you are correct.

Beyond this example, are "inverse lensing" effects generally not possible in GR, or are they technically possible? Put in another way, is it possible in GR that space could be littered with "wide angle" lenses, so to speak, or is that inherently incompatible with GR?

Last edited: Sep 30, 2016
5. Sep 30, 2016

### GeorgeDishman

The Planck Mission measured the curvature and found it to be close to flat, $Ω_K=0.000±0.005$ which means that in principal objects could be farther or nearer than we think but only by something around 1%, probably much less, and certainly not "orders of magnitude".

My figures come from section 6.2.4 and in particular eqn(50) of the Planck 2015 Results.

6. Sep 30, 2016

### Staff: Mentor

If by "inverse lensing" you mean an isolated object that would cause a lensing effect opposite to the kind that normal isolated massive bodies produce, then no, I don't think such objects are realistically possible in GR. Technically, yes, you could construct a mathematical solution that had this property--it would be an isolated object with negative mass, which would reverse the signs of the curvatures produced in the vacuum region outside the object (so the tangential curvature, the one that produces the usual lensing effects, would be negative instead of positive, and the radial curvature would be positive instead of negative). But such an object could not hold itself together in a bound state; it would explode, because the gravity between the parts of the object's interior would be repulsive, not attractive. (Also, of course, there is the fact that no known material has this property to begin with.)

7. Sep 30, 2016

### particlezoo

The conclusion that the curvature of the universe is close to flat comes from the observation of the uniformity of the cosmic background radiation. Rapid early expansion of the universe in the context of the Big Bang model would imply such uniformity of the cosmic background radiation. Simultaneously, such rapid early expansion of the universe implies that the curvature of the universe is close to flat. So when the cosmic background radiation is seen to have a certain level of uniformity, it is implied (abductively) that the rate of expansion was fast enough such that the curvature of the universe is therefore close to flat.

Strictly speaking, the uniformity of the cosmic background radiation requires parts of the universe to be very close to each other early on for a long enough period of time to iron out fluctuations in temperature. The common explanation is that there was a period of relatively slow expansion before a subsequent inflationary period with a very high rate of expansion.

However, doesn't the common explanation implicitly assume that different parts of the universe could not have been connected by some other means, such as regions of hyperbolic space connecting otherwise too distant parts of the universe? Couldn't these regions then have been due to inhomegenities in the distribution of dark energy during the early universe? If so, isn't it possible then that the dark energy hasn't actually evened out and what we are seeing out there has been from the beginning a much smaller "universe" than what it appears to be, allowing the cosmic background radiation to be uniform without the curvature of the universe being close to flat?

Sincerely,

Kevin M.

P.S. for reference:

http://abyss.uoregon.edu/~js/21st_century_science/lectures/lec21.html

Last edited: Sep 30, 2016
8. Sep 30, 2016

### Staff: Mentor

First, the curvature you are talking about is spatial curvature, not spacetime curvature. You should be thinking about spacetime curvature. Spatial curvature is observer-dependent; spacetime curvature is not.

Second, the uniformity of the CMBR is strong evidence for isotropy, but isotropy is not the same thing as spatial flatness; the isotropy of the CMBR, by itself, tells us nothing about the universe's spatial curvature, since there are possible models which are isotropic that have all three possible kinds of spatial curvature--positive, negative, and zero.

The conclusion that the universe is spatially flat (more precisely, that it is spatially flat as seen by comoving observers, observers who see the universe as homogeneous and isotropic--observers in different states of motion won't see the universe as spatially flat) is based on comparing the actual average density of the universe as a whole, with the "critical density" that indicates spatial flatness. Current measurements tell us that the actual density is the same as critical to within our measurement accuracy. See Ned Wright's cosmology tutorial here:

http://www.astro.ucla.edu/~wright/cosmo_03.htm

Yes.

No, this is backwards. The common explanation is that there was an initial period of inflation, very rapid accelerating expansion, and then relatively slow expansion from the end of inflation on.

9. Sep 30, 2016

### particlezoo

So you are essentially saying that we observe the curvature of the space of the universe to be flat because the actual density is close to the "critical density" necessary for the curvature of the universe to be flat. But I thought dark energy (which if I'm not mistaken partially contributes the "actual density") has since been postulated to explain more recent observations that suggest within the current framework that the expansion of the universe is accelerating. Since, as you say, the average density as a whole is what indicates the overall flatness of the universe's space curvature, wouldn't it mean that the "actual density" was calculated from the prior inference that the universe's space curvature was flat?

That page is rather old, and according to the page, it was last updated July 3, 2009, which happens to be the same day that the Planck satellite entered service.

Here is the where this article differs from its prior version. On the current version we see (from July 3, 2009):

From a previous version of the webpage on May 30, 2009, it reads:

I can't think of any physical measurements that could be anywhere close to this level of accuracy. From this information alone, I am quite sure that the "actual density" was inferred (i.e. back calculated) from a pre-existing boundary condition.

http://www.ctc.cam.ac.uk/outreach/origins/inflation_zero.php

Last edited: Oct 1, 2016
10. Oct 1, 2016

### Staff: Mentor

Yes, it does. Dark energy density is about 70% of critical; dark matter density is about 25% of critical; and ordinary matter density is about 5% of critical. Adding them all together gives 100% of critical.

The number for the dark energy density is calculated from the rate of acceleration of expansion, yes.

No. We didn't know the universe was spatially flat before. We derived that as a consequence after we had reasonably good estimates for the density, including dark energy density.

The numbers Ned Wright is quoting are theoretical numbers, not measured numbers; they are what would have to be true at the very earliest times in order for the universe to be spatially flat now to the accuracy we actually measure, if our current theoretical models are correct. The obvious conclusion is that our current theoretical models are not quite correct, or at least they are incomplete since they give no explanation of how the universe's initial conditions could have been fine tuned to that level of accuracy. But the actual measurements that show our universe being spatially flat now, to a given level of accuracy, are logically prior to all of this.

You are incorrect. The problem Ned Wright is describing in what you refer to is a theoretical problem, not a measurement problem. The problem is not that we have to infer the actual density from a boundary condition that the universe must be spatially flat. The problem is that our actual measurements tell us that the universe is spatially flat, but theoretically--if our current theoretical models are correct--it's hard to understand how that can be true this long after the Big Bang, given that the expansion was not accelerating during much of that time (it was accelerating rapidly for a very brief period during inflation, then was decelerating for billions of years, then only started accelerating again, slowly, within the past couple of billion years or so), and given that, if the expansion is not accelerating, the universe should become less spatially flat over time.

None of this has anything to do with spatial flatness. It has to do with isotropy--how can the CMBR be as isotropic as it is today, if the portions of it that we see coming from opposite regions of the sky were never in causal contact? Notice that the website describes this as the "horizon problem", not the "flatness problem"; the latter is the problem I described above, and is a different problem from the problem of how the CMBR can be as isotropic as it is.

11. Oct 1, 2016

### GeorgeDishman

Both statements are correct, the common (but probably outdated) explanation is that inflation swelled a small region which had enough time to reach equilibrium to a size much greater than our horizon hence we see a nearly uniform temperature, which implies three periods of expansion, first slow then fast (inflation) then slow. That seemed to be in conflict with what I've seen more recently which said all matter was produced at the end of inflation hence could not have reached equilibrium before inflation started, which is why I asked this question:

My view of inflation and reheating come from sources like these but it's an area I have only started to skim through:

http://cds.cern.ch/record/346052/files/9802221.pdf

http://www.damtp.cam.ac.uk/user/nb465/Talks/reheating-rev.pdf

https://ned.ipac.caltech.edu/level5/Liddle/Liddle5_6.html

Regarding the measurement of flatness, the old WMAP site has a "Build a Universe" page which made it look as though the flatness could be derived from the angular power spectrum rather than isotropy (although finding the density might be an intermediate calculation).

http://map.gsfc.nasa.gov/resources/camb_tool/

As I said, my figures came from the Planck results. In that, while CMB measurements gave a somewhat positive curvature, there is a significant degeneracy in using that alone and including BAO is necessary to refine the result to the accuracy quoted. See section 6.2.4:

Planck 2015.

Since BAO act as a "standard ruler", my impression is that they directly address the original question regarding any possible error in distances which might result from a non-zero curvature.

12. Oct 1, 2016

### Staff: Mentor

Why does it imply a first period of slow expansion? Our current models don't really say what happened before inflation. AFAIK there is no claim that our current data implies that a period of slow expansion must have preceded inflation.

The isotropy of the universe is not limited to matter (or the CMBR). It also includes isotropy of the spacetime geometry and of all the quantum fields involved. The "wrinkles" in the CMBR that have been observed by WMAP and more recently Planck are evidence of small variations in spacetime geometry and quantum fields; those existed during inflation, not just after it.

The "reheating" at the end of inflation did not "produce matter" in the sense of bringing into existence something that didn't exist before. What it did was transfer a large energy density from the inflaton field to the fields of the Standard Model (leptons, quarks, gauge bosons). But the SM fields already existed; they just had negligible energy density prior to the end of inflation.

13. Oct 2, 2016

### GeorgeDishman

I understand that the fields per se would have already existed but I'm not clear on the distinction you are making, I would have thought that putting energy into the field had the effect of "creating" particles in the classical sense? What you say seems to be at odds with the sources I quoted, let me include some quotes to show why I am confused:

http://cds.cern.ch/record/346052/files/9802221.pdf

"In modern versions of inflationary cosmology there is no pre-inflationary hot stage, the Universe initially expands quasi-exponentially in a vacuum-like state without temperature. During inflation, all energy is contained in the inflaton field $Φ$ which is slowly rolling down to the minimum of its effective potential $V(Φ)$. .. In this scenario all the particles constituting the Universe are created due to the interaction with the oscillating inflaton field. Gradually, the energy of inflaton oscillations is transferred into energy of the ultra-relativistic particles. Eventually created particles come to a state of thermal equilibrium at some temperature $T_r$, which is called the reheating temperature.
To describe creation of elementary particles from the inflaton oscillations, we shall consider the interaction terms in the fundamental Lagrangian."​

http://www.damtp.cam.ac.uk/user/nb465/Talks/reheating-rev.pdf

"Conclusions
• Reheating is a key part of the inflationary story.
◮ Necessary to recover hot BB, explains the origin of all particles in the universe.​
• Multi-stage process: explosive particle production, rescattering, fragmentation, turbulence, thermalization."
https://ned.ipac.caltech.edu/level5/Liddle/Liddle5_6.html

"During inflation, all matter except the scalar field (usually called the inflaton) is redshifted to extremely low densities. Reheating is the process whereby the inflaton's energy density is converted back into conventional matter after inflation, re-entering the standard big bang theory."​

14. Oct 2, 2016

### Staff: Mentor

I'm not sure what you mean by "in the classical sense". What you are calling "putting energy into the field" is just a transition in the field from one state to another. Labeling the initial state of the field as "vacuum" and the final state as "containing particles" and calling the transition "particle creation" is just terminology, not physics. The physics is that you have a bunch of quantum fields that undergo state transitions--at the end of inflation the inflaton field transitions from a high energy to a low energy state, and the Standard Model fields transition from a low energy to a high energy state.

To turn your question around: why do you think "creation" is such a big deal? Why is it so important to pin down whether "creation in the classical sense" is taking place? What physical (as opposed to terminological) difference does it make?

15. Oct 4, 2016

### GeorgeDishman

Well in simple terms, I am made of neutrons, protons (which in turn are composed of quarks, etc.) and electrons so if those particles had never been created, I would not exist. That, for me personally, would be a 'big deal'. In terms of physics, the details of baryogenesis are a topic of considerable interest.

16. Oct 4, 2016

### Staff: Mentor

In other words, if inflation had not ended and the quantum field state transition I described had not happened, you would not exist. That's true. What does it have to do with the topic of this thread?

That's fine, but what does it have to do with physics? Or the topic of this thread?

Yes, they are. Once again, what does that have to do with the topic of this thread? You started out by saying there appeared to be a conflict between "matter being created at the end of inflation" and "thermal equilibrium existing before/during inflation". I have shown that there is no such conflict. None of your assertions quoted above have anything to do with that.

17. Oct 4, 2016

### GeorgeDishman

Very little, I was simply answering your question "why do you think "creation" is such a big deal?".

18. Oct 4, 2016

### Staff: Mentor

You didn't take into account the key qualifier I added:

None of the things you bring up make any physical difference. Even with respect to baryogenesis, nobody studying the subject cares, as far as physics is concerned, whether "creation in the classical sense" took place. That has nothing to do with the physics involved; it's just a question of terminology.

19. Oct 4, 2016

### GeorgeDishman

OK, thanks for that. I participate here in order to learn more and I'm very much a beginner so clarifying my misunderstanding of what I have read is always a help.