# Centre of the universe?

by yuiop
Tags: centre, universe
P: 28
[/I]
 Quote by kev I have seen the balloon analogy which represents an unbounded surface to a 2 dimensional creature. That creature would assume a flat surface and no matter how far it kept going in one direction, it would never find an edge. However, if the balloon surface was tiled and it marked those tiles it would eventually work out that it was going round in circles and that the number of tiles (galaxies) was finite. In a typical description of the evolution of the universe, it is explained that the expansion of the universe is accelerating "now", but in an earlier epoch the expansion was slowing down when gravity dominated the cosmological constant. The problem I have is that if the universe is infinite with an infinite amount of mass..... Kev, you seem to be mixing 2 scenarios here. The balloon analogy is not an infinite universe. The surface area can be calculated as a specific number, not infinity. Maybe you meant unbounded (no edge) ....., then that slow down would have been impossible because every massive body would equally attracted in all directions by an equal amount of mass in all directions. There would be no preferred direction for any body to move in. Well yes, I think so ish but deviation from homogenous gives an opportunity for clumping together I agree this is idiosyncratic view, but I just have difficulty imagining an infinite amount of mass. [I]I think for infinte extent and mass you need a different analogy to the balloon. Consider a sponge cake with raisins (galaxies) placed in the oven. The cake 'rises' expands (space expands) and all the raisins move away from each other. Imagine this 'going on forever' in all directions. No-one can imagine this sort of infinity in it's completeness!! /I] The closest thing I can imagine to the concept of an infinite universe is one expanding at the speed of light making it impossible for any observer to travel fast enough to get to the “edge” of that universe. However, if the expansion of the universe is due to the expansion of space itself, then the apparent superluminal recession of the most distant galaxies is not an insurmountable problem because any traveller would be moving relative to the local space and the traveller would eventually appear to have superluminal velocity relative to a stay at home observer. The 2D surface of the balloon wraps around on itself in a 3D universe that is not visible to the 2D inhabitants of the balloon surface. Now you are talking finiteThe analogy implies our universe is a 3D space that wraps around on itself in a 4D space that is not visible to us. I believe some researchers have tried to find evidence of this wrapping around effect by looking for symmetries in the pattern of distant galaxies in different directions at the edge of the visible universe and never found that evidence. Yes I find it odd that they look as it seems clear that we can see only a part of the universe and this means probably the major proportion is receeding at >c ; hardly a situation where you could see the wrap around Another difficulty I am having is how the universe started everywhere at the same time. Assuming that an initially finite universe cannot turn into an infinite universe later on, then if the universe is infinite now, it must have been infinite at the start. I think this is something where you have to ask 'what infinity am I talking about?' My point is that the universe had a beginning in time.... Presumably you are referring to spatial infinity and spatial expansion. So, with a spatially infinite universe with a start time imagine your flat surface with dots on it. Each dot expands to produce an expanding space. Now imagine, on that flat surface an infinite number of dots (well separated by a plank unit say) each being the Big Bang for the space that the dot expands to.Lol. But basically that's valid. A start time and infinite space. Ah reread your post. Yes I think I agree, a spatial infinity suggests an spatially infinite start.... not sure... How would the start of initially infinite universe be synchronised so that it starts everywhere at the same time? Any synchronisation could not be coordinated by signals constrained to the speed of light. The best explanation I have seen is that the universe was kick started by the collision of two “branes” The 2D analogy would be two parallel sheets colliding, making the collision and start of the universe happen simultaneously everywhere on the surface of those sheets. The 3D version would two infinite 3D branes floating around in God knows what, before colliding and starting our universe. Now all we have to do is figure out how the infinite 3D branes were created.
I think I am correct to say that in modelling these Branes it is assumed slight imperfections means some areas contact and others do not. I might be out of date there though. Remember these branes are multidimensional and the flat surface image is obtained by dumping a number of dimensions.
 P: 3,967 Muccasen, thanks for replying to my controversial post in such a pleasant manner. I have to pop out for a while, but would like to continue this discussion later.
Astronomy
PF Gold
P: 23,271
kev, this is good post, with points that are interesting to reply to

I see Muccasen already responded, and also that you said you had to go and would be away for a while. But I will reply to one or two things immediately.

 Quote by kev I have seen the balloon analogy which represents an unbounded surface to a 2 dimensional creature. That creature would assume a flat surface and no matter how far it kept going in one direction, it would neverfind an edge. However, if the balloon surface was tiled and it marked those tiles it would eventually work out that it was going round in circles and that the number of tiles (galaxies) was finite.
I am glad you are familiar with that "nearly flat" slight positive curvature alternative to the infinite case. Recently cosmology papers are using phrases like flat or nearly flat. The majority still think spatial infinite (with mass infinite of course) but they increasingly allow this S3 finite volume case as a possibility.

I see you have a conceptual problem with the INFINITE case:

 In a typical description of the evolution of the universe, it is explained that the expansion of the universe is accelerating "now", but in an earlier epoch the expansion was slowing down when gravity dominated the cosmological constant. The problem I have is that if the universe is infinite with an infinite amount of mass, then that slow down would have been impossible because every massive body would equally attracted in all directions by an equal amount of mass in all directions. There would be no preferred direction for any body to move in.
This is a keen, insightful objection, but it fails because of General Relativity. So far GR is dominant as the theoretical basis of Cosmology because it predicts successfully to high accuracy the way no other theory of gravity does. Cosmology is based on a highly simplified model called the Friedmann equation which is derived from the Einstein GR field equation. It is one particular solution of GR. In the Friedmann model distances are either all increasing or all decreasing. With the parameters of our particular universe, as measured to date, we expect continued increase (no eventual collapse).

In the Friedmann model there IS NO PREFERRED DIRECTION to the expansion, and no preferred direction to any acceleration or deceleration of expansion. Expansion does NOT CONSIST OF GALAXIES MOVING THRU SPACE.
Expansion is just a percentage increase in the distance between stationary points.

GR teaches us to expect distance to change unless it is a distance locked in by atomic or other forces----like the crystal bonds of a metal rod, or the orbit dynamics of a planetary system. Within certain bound structures, forces keep distances from changing, but otherwise one should expect them to be expanding or contracting. The Friedmann equation is beautifully simple and you can see immediately how it governs the rate of expansion/contraction. It works either in the finite volume or infinite volume case and (as I said) doesn't need any force vector in any preferred direction. There's just this timedependent scalefactor a(t) and an ordinary differential equation that tells about the time derivative a'(t) and how it is determined. It is probably easier to understand in that form than in the form of words.

Currently the percentage rate of expansion or more exactly the fractional rate (which you could write a'(t)/a(t) if you want) is one percent every 140 million years. You can see why it has to be. The left hand side of the Friedmann equation is actually the square of that very quantity----(a'(t)/a(t))2.
Actually there are two Friedmann equations, both are simple and you should look at both. But for some reason everybody always talks about THE equation as if there were just one. the other one is about the second derivative of the scalefactor: a''(t).

when you look at these two Friedmann equations the first thing you see is that they work for both the infinite volume case and the positive curved S3 case. that is made explicit with a parameter k which can be either +1, 0, or -1

You are right that expansion, as far as observations can tell us by fitting the data, was DECELERATING until some 4 or 5 billion years ago! That was when matter was denser and its effect prevailed over the cosmological constant. And then with expansion matter thinned out and the cosmological constant effect of acceleration began to prevail.
But all this stuff works equally well in the Friedmann model whether you pick the infinite case or the finite case!. This doesn't need any preferred direction. And the expansion is not outwards from any particular point. Basically it is just GR effects involving the dynamics by which distances change (GR's dynamic geometry).

 I agree this is idiosyncratic view, but I just have difficulty imagining an infinite amount of mass.
Well, in that case just picture space as S3! A large balloon is almost flat. This fits the data even slightly better than the infinite space model-----it is just that the infinite flat version is mathematically simpler and the difference in fit is not statistically significant at this point.

 I believe some researchers have tried to find evidence of this wrapping around effect by looking for symmetries in the pattern of distant galaxies in different directions at the edge of the visible universe and never found that evidence.
That is right. I mentioned some of that research----papers by Neil Cornish and David Spergel----in an earlier post in this thread. They did not rule out the finite picture. They just proved that if it was finite it had to be BIG.

But that is what the curvature data already suggested anyway.

IIRC a 2007 Ned Wright paper presented a good fit for Omega as 1.011, and in a landmark 2006 paper by Spergel et al the errorbar all on the upside of 1------something like [1.010, 1.041] at 68 percent confidence. Ned Wright's 1.011 falls in that range.
And that 1.011 would indicate an S3 circumference of some 800 billion LY, which was way more than the lowerbound that Cornish and Spergel came up with in 2004 (and later refined in 2006). So that much agrees.

====================

What I have still not given you is INTUITION for why an infinite flat volume that starts off expanding should ever slow down! The real reason is that GR is the most accurate theory of gravity at large scale that we have and if you buy GR then you buy Friedmann derived from it and you buy that in the interesting cases expansion slows down unless there is a cosmological constant. But this is unsatisfying. One still wants some intuition for why in the hell it happens.

The puzzle is in the infinite flat case. There is no preferred direction, the net effect of everything should be zero---why should expansion be slowed down? I hope someone else has some intuition about this. When I think about it I come up short on the intuition end.

One thing to remember is that GR is a field theory and information only travels at a finite speed. So in some big region the local patch of geometry doesnt KNOW about the matter way out there. the effective distribution of matter in its past light cone could be very different from the instantaneous (approximately even) distribution. But I'm not sure enough about this. the problem of how you get intuition for the slowing down is interesting----maybe there is some obvious answer that i'm missing. We might hear from SpaceTiger or Garth about this.
 P: 28 Marcus:"lots more to say but I too have to go out" We look forward to yr return!!
 P: 96 If the universe is homogeneous on the largest scales, then shouldn't other observer's red shifts produce at least an equal amount of blue shifts to red shifts for us? That assumption surely has not been verified. Why did Hubble and so many other scientists reject a nearby center when observations say otherwise?
 P: 96 Thanks for the balloon explanation, but hear is the problem that I have with that model: If space is expanding, then aren't lengths themselves expanding? What is the expansion relative too then? What I see happening are masses moving away from each other, nothing more. I think historically this has been interpreted as space expansion to fit the large scale homogeneity model.
 P: 96 *here
P: 15,319
 Quote by lewis198 If space is expanding, then aren't lengths themselves expanding?
Just as in the balloon analogy, the expansion does not stretch physical objects. A ruler glued to the surface of the balloon does not magically get longer.
P: 3,967
Thanks to Marcus, WFWINTERS and mucassen for all the interesting and informative responses. There are a lot issues to address here so I hope you don’t mind if I reply to just a few paraphrased points while I try and gather my thoughts. :P

 Quote by WFWINTERS Lewis198, To help give a visual of how every point has equal footing on saying it's the center, based on redshift, picture this. Using the example stated previously of 3D space being analogous to the 2D surface of a sphere, let's further imagine that this sphere is a balloon. If you were to plot many equally spaced points on the surface of that spherical balloon, you would find that as you inflated the balloon and uniformly increased it's surface area, any one of those points would see all of the other points moving away from it. An observer at any point on the surface of an inflating sphere can confidently say that every other point it sees is retreating from their location, giving that observer the illusion that they are at a special central location. This is an interesting effect because it implies that any point in the universe would give an observer the impression that they are at the origin of the universe's expansion, and in a way, all of those observers would be correct.
I don’t have any trouble with concept that any 2D observer on the surface of an expanding balloon would appear to be at the centre of expansion of their own little visible universe. Neither do I have any trouble with a 3D observer in an expanding raisin cake appearing at to be at the centre of expansion, no matter where they are located within the raisin cake. The problem I am having is imagining a spatially infinite raisin cake so that no observer finds themselves at the edge of the raisin cake. I used the balloon example as it illustrates a surface with no edge or boundary (same thing?) while still having a finite surface area. When people say the universe has no edge or boundary, do they mean it is just an illusion like the surface of the balloon, where infinite distance is merely achieved by going round in circles? For example, if some small creatures at the same starting point went in 4 different directions away from each other they would eventually bump into each other on the other side of the balloon. Does this imply that if we scale up the 3D balloon surface to a 3D space embedded in 4 dimensions, that travellers going in opposite directions in a straight line eventually meet each other, and is that how our universe is modelled?

Going back to Muccasen’s example of the raisin cake, what would an observer on the edge of an expanding but finite cake see? Light coming from raisins on the surface of the cake to either side of him would curve inwards towards the mass at the centre of the cake. The photons arriving at his location would appear to be coming from outside the cake so maybe even observes on the edge would not know they are on the edge unless they do some very careful measurements? This is similar to the aberration seen by a relativistic traveller that makes light from objects to the side of him appear to be coming from somewhere in front of him.

 Quote by muccasen Kev, you seem to be mixing 2 scenarios here. The balloon analogy is not an infinite universe. The surface area can be calculated as a specific number, not infinity. Maybe you meant unbounded (no edge)
Yes, I meant unbounded.

 Quote by muccasen The 2D surface of the balloon wraps around on itself in a 3D universe that is not visible to the 2D inhabitants of the balloon surface. Now you are talking finite.
Yes, I am talking finite. I am trying to get to the crux of the matter when people talk about an infinite universe do they mean really infinite in the full horror of the true meaning of infinite or do they mean an illusion of infinite brought about by 3D space wrapping around on itself just as the 2D surface of the sphere wraps round on itself?

 Quote by muccasen Well yes, I think so ish but deviation from homogenous gives an opportunity for clumping together
We see the clumping together at quite large scales, for example the acceleration of all the galaxies towards the “great attractor” in our local group. We tend to ignore or correct for this local clumping when considering the cosmology of the universe in general.

 Quote by muccasen I think this is something where you have to ask 'what infinity am I talking about?'
I was thinking more in terms of spatial infinity rather than infinite time.

I have read the ideas of Cantor on infinity and I can see how they apply to intangible things like real and imaginary numbers but when applied to physical objects like atoms or galaxies I have a problem. A spatially infinite universe with an even distribution of infinite mass gives means the universe is like a real incarnation of the Hilbert Hotel and what a messy thing that is! See http://www.ccs3.lanl.gov/mega-math/w...y/inhotel.html

 Quote by marcus I am glad you are familiar with that "nearly flat" slight positive curvature alternative to the infinite case. Recently cosmology papers are using phrases like flat or nearly flat. The majority still think spatial infinite (with mass infinite of course) but they increasingly allow this S3 finite volume case as a possibility. . . IIRC a 2007 Ned Wright paper presented a good fit for Omega as 1.011, and in a landmark 2006 paper by Spergel et al the errorbar all on the upside of 1------something like [1.010, 1.041] at 68 percent confidence. Ned Wright's 1.011 falls in that range. http://physicsforums.com/showthread....02#post1558402 And that 1.011 would indicate an S3 circumference of some 800 billion LY, which was way more than the lowerbound that Cornish and Spergel came up with in 2004 (and later refined in 2006). So that much agrees.
Wikipedia defines S3 as 3-dimensional spherical geometry with a small (positive) curvature, often notated as S3 and Omega as the average density of the universe divided by the critical energy density, i.e. that required for the universe to be flat (zero curvature).

Since mass density is a ratio of mass to volume it does not tell us if the mass or volume is infinite or finite. Omega is a ratio of mass density to critical energy density so I’m not sure if Omega tells us anything about whether the volume or mass if infinite or finite. The wikepedia article seems to imply the observed Omega>1 implies a finite radius (and finite mass?) for the universe.
Thanks for pointing out the Friedmann equations to me. They look interesting but they are new to me and I will have to spend some time trying to understand them.

 Quote by marcus One thing to remember is that GR is a field theory and information only travels at a finite speed. So in some big region the local patch of geometry doesnt KNOW about the matter way out there. the effective distribution of matter in its past light cone could be very different from the instantaneous (approximately even) distribution. But I'm not sure enough about this. the problem of how you get intuition for the slowing down is interesting----maybe there is some obvious answer that i'm missing. We might hear from SpaceTiger or Garth about this.
I am glad you agree that there is an intuition problem with gravity in the infinite mass case and yes, it would be interesting to see what others think. As far as the past light cone is concerned my view is this. Say A is only influenced by the mass within his visible universe (which would be a consequence of the influence of gravity being limited to the speed of light) and if mass is evenly distributed within his visible universe then A is effectively at the centre of gravitational mass and is not attracted in any direction. (The visible universe from A point of view is the past light cone of A). Say B is on the edge of A’s visible universe. B will have his own visible universe and B will also be at the centre of his own visible universe and effectively at the centre of mass within his own past light cone. There is no particular reason for B to be gravitationally attracted to A or vice versa.
P: 15,319
 Quote by kev The problem I am having is imagining a spatially infinite raisin cake so that no observer finds themselves at the edge of the raisin cake. I used the balloon example as it illustrates a surface with no edge or boundary (same thing?) while still having a finite surface area. When people say the universe has no edge or boundary, do they mean it is just an illusion like the surface of the balloon, where infinite distance is merely achieved by going round in circles?
It's not an illusion. Though I see where you're going with that.

Having no edge or boundary does NOT automatically imply infinite distances. That is the point of the balloon analogy.

 Quote by kev For example, if some small creatures at the same starting point went in 4 different directions away from each other they would eventually bump into each other on the other side of the balloon. Does this imply that if we scale up the 3D balloon surface to a 3D space embedded in 4 dimensions, that travellers going in opposite directions in a straight line eventually meet each other, and is that how our universe is modelled?
Yes.
P: 3,967
 Quote by kev .... Does this imply that if we scale up the 3D balloon surface to a 3D space embedded in 4 dimensions, that travellers going in opposite directions in a straight line eventually meet each other, and is that how our universe is modelled?
Oops..typo.. I meant Does this imply that if we scale up the 2D balloon surface to a 3D space embedded in 4 dimensions, that travellers going in opposite directions in a straight line eventually meet each other, and is that how our universe is modelled?
Astronomy
PF Gold
P: 23,271
 Quote by kev ...Does this imply that if we scale up the 2D balloon surface to a 3D space embedded in 4 dimensions, that travellers going in opposite directions in a straight line eventually meet each other, and is that how our universe is modelled?
It is how it is modeled by some people.

You have to make allowances for bumpiness. On a perfect (not bumpy, not expanding) sphere they will meet. But in the real world, stars and galaxies deform the space around them so that geodesics are deflected. If the two travelers are not prevented by accelerating expansion from ever reaching each other then they will approximately meet.

You ask:...and is that how our universe is modelled?

Indeed in Quantum Cosmology some of the best modeling work that I know of uses spacetimes where the spatial sections are approximately S3-----that is they are topological three-spheres but they have the bumps and wrinkles you would naturally expect

Amjorn and Loll's computer models are all of that type. they run sims of quantum universes that come into existence, grow, shrink, and disappear according to some simple rules. They use S3 spatial sections. If you want links, ask.

Ashtekar group's computer models, many of them, are also of that type. that's where they replace the cosmological singularity with a bounce----a prior universe which is spatially a "bumpy" threesphere collapses to Planck density and re-expands. They use S3 spatial sections. If you want links, ask.

these are experimental quantum cosmology modeling----I'm not saying the real universe spatial topology is S3 because we don't know---I'm saying what some leading model-builders do.

You asked is it MODELED like that and the answer is certainly YES.

Indeed the dominant classical model that working cosmologists use is LCDM and it comes mostly in two flavors as I said----Omega=1 flat infinite and Omega>1 finite with S3 spatial sections. So you see that kind of model whether you look at classical cosmology or the newer quantum cosmology approaches (especially since 2004 or 2005)
Astronomy
PF Gold
P: 23,271
 Quote by kev ...Yes, I am talking finite. I am trying to get to the crux of the matter when people talk about an infinite universe do they mean really infinite in the full horror of the true meaning of infinite or do they mean an illusion of infinite brought about by 3D space wrapping around on itself just as the 2D surface of the sphere wraps round on itself?
when they say infinite they mean really infinite.
spatial infinite means infinite spatial volume.
it is mathematically convenient although it gives people philosophical vertigo

 Since mass density is a ratio of mass to volume it does not tell us if the mass or volume is infinite or finite. Omega is a ratio of mass density to critical energy density so I’m not sure if Omega tells us anything about whether the volume or mass if infinite or finite. The wikepedia article seems to imply the observed Omega>1 implies a finite radius (and finite mass?) for the universe. ...
Set your mind at rest. That is exactly what Omega tells us about. what you say the Wikipedia seems to imply is correct.

the critical energy density is the density that must be surpassed in order to force the solution of Einstein/Friedmann equation to have finite spatial volume.
So naturally since Omega is the RATIO, if Omega > 1 that means precisely that the observed density DOES surpass critical and therefore we must have finite spatial volume.

aka positive curvature (one implies the other)

I'm glad you are interested in these things. I find I am having to repeat myself a fair amount. What I have been telling you is manifest in the Friedmann equations, so they might be something good for you to study. It would help resolve your doubts perhaps.
 P: 3,967 Marcus, I wouldn't mind seeing those links you mentioned, just out of interest. Can't promise I'll understand them ;)
Astronomy
PF Gold
P: 23,271
 Quote by kev Marcus, I wouldn't mind seeing those links you mentioned,..
Sure, for Ambjorn and Loll the two recent papers are
want to join me in studying up on CDT in early 2008 then the articles to print off so you have hardcopy to scribble are:

http://arxiv.org/abs/0711.0273 (21 pages)
The Emergence of Spacetime, or, Quantum Gravity on Your Desktop

"Is there an approach to quantum gravity which is conceptually simple, relies on very few fundamental physical principles and ingredients, emphasizes geometric (as opposed to algebraic) properties, comes with a definite numerical approximation scheme, and produces robust results, which go beyond showing mere internal consistency of the formalism? The answer is a resounding yes: it is the attempt to construct a nonperturbative theory of quantum gravity, valid on all scales, with the technique of so-called Causal Dynamical Triangulations. Despite its conceptual simplicity, the results obtained up to now are far from trivial. Most remarkable at this stage is perhaps the fully dynamical emergence of a classical background (and solution to the Einstein equations) from a nonperturbative sum over geometries, without putting in any preferred geometric background at the outset. In addition, there is concrete evidence for the presence of a fractal spacetime foam on Planckian distance scales. The availability of a computational framework provides built-in reality checks of the approach, whose importance can hardly be overestimated."

http://arxiv.org/abs/0712.2485 (10 pages)
Planckian Birth of the Quantum de Sitter Universe

"We show that the quantum universe emerging from a nonperturbative, Lorentzian sum-over-geometries can be described with high accuracy by a four-dimensional de Sitter spacetime. By a scaling analysis involving Newton's constant, we establish that the linear size of the quantum universes under study is in between 17 and 28 Planck lengths. Somewhat surprisingly, the measured quantum fluctuations around the de Sitter universe in this regime are to good approximation still describable semiclassically. The numerical evidence presented comes from a regularization of quantum gravity in terms of causal dynamical triangulations."

 Quote by marcus Ambjorn and Loll's computer models are all of that type. they run sims of quantum universes that come into existence, grow, shrink, and disappear according to some simple rules. They use S3 spatial sections. ... Ashtekar group's computer models, many of them, are also of that type. that's where they replace the cosmological singularity with a bounce----a prior universe which is spatially a "bumpy" threesphere collapses to Planck density and re-expands. They use S3 spatial sections. ... these are experimental quantum cosmology modeling----I'm not saying the real universe spatial topology is S3 because we don't know---I'm saying what some leading model-builders do.
I'll get a link to some Ashtekar et al work.
http://arxiv.org/abs/gr-qc/0612104
Loop quantum cosmology of k=1 FRW models
Abhay Ashtekar et al
(Submitted on 18 Dec 2006)

In this paper they do not put in the effect of a postive cosmological constant, so the universes eventually collapse. that wouldn't have to happen----you could have a contracting phase, a bounce, and then an expanding phase that went on expanding indefinitely. k=1 means the case with space looking like S3
FRW means Friedmann Robertson Walker----the prevailing model in cosmology which has various flavors depending on what parameters you put in.

With technical papers there is usually an understandable non-technical introduction section and conclusions section. Not to worry about understanding the whole paper. Just get what you can out of it. And this is just to get a sample of people doing computer models
P: 1
Hi sorry for barging in. Any idea if the Universe has at least a 2 dimensional geometry??

 Quote by DaveC426913 No, the universe is not infinite. If you are being told this, seek a more informed source for your information. By current estimates the universe is about 156 billion light years wide.
P: 28
 Quote by dipayankar Hi sorry for barging in. Any idea if the Universe has at least a 2 dimensional geometry??
You Kidding? Current thinking is 3 spatial 1 time and a number of curled up ones totalling 11 or so
P: 28
 Quote by muccasen You Kidding? Current thinking is 3 spatial 1 time and a number of curled up ones totalling 11 or so
But I favour 4 macro spatial + Loads o other macros not relevant to us but take no notice of me on that 1 !

 Related Discussions Astronomy & Astrophysics 15 Astronomy & Astrophysics 22 Astronomy & Astrophysics 37 Astronomy & Astrophysics 7 Astronomy & Astrophysics 58