Wednesday, August 10, 2016

Map of the Universe

 Sean Carroll's excellent From Eternity to Here inspired me to create a map of the universe that capture some of the essential physics. I used Photoshop to assemble the collage below from the components described underneath. It shows a timeline of the universe at left and the observable universe at right, linked by a faint light cone. (click on it to expand).

The ingredients of the collage are:

  • Timeline of the Universe, from NASA's WMAP project, which mapped the cosmic microwave background fluctuations that appear as the blue/green pattern at the left (near the big bang) and as the outer surface of the observable universe / sky map on the right. The original diagram below shows stars and galaxies in the appropriate epochs, but it oversimplifies in some ways. Since the diagram shows two dimensions of space and one of time, the stars and galaxies should be world lines, or tubes, along the time dimension. I smeared some of them with Photoshop to suggest this, but I was not able to generate "realistic" tubes. Mine look vaguely comet-like; they are far too short, and they have beginnings and ends, which would imply the sudden appearance or disappearance of stars or galaxies. They should all begin at the Big Bang and end at the right edge of the funnel, and they should branch, merge and change shape over history. To properly represent, say, the collapse of a dust cloud to form a star, a wide cloud tube would have to narrow and turn into, or branch off to form, a star tube. The mutual recession of the galaxies detected by Hubble should be represented as a flowerlike divergence of galaxy-tubes. Finally, as far as I can tell the stars and galaxies in the original image did not represent any particular real stars and galaxies, and so cannot be put into any detailed correspondence with the stars and galaxies inside the sky sphere at right. Ideally you would want the Earth-tube to be on the central axis of the funnel. It would take a computer to generate a realistic diagram like this with tubes based on real objects. You might need to figure out a way to keep the tubes from obscuring each other, if they are dense enough -- perhaps by making them semitransparent. Also since this is a 2D slice evolving through time you would have to choose some 3D (2 space x 1 time) slice of the real 4D history...

  • The sphere on the right of the collage, created by the Deus Consortium,  depicts the observable universe (a.k.a. "the sky") as a sphere, with the cosmic microwave background on its outer surface (since this is the oldest/most distant light we can observe). An octant of the sphere is cut away, showing axes labeled with time and red-shift (z). Sadly the interior is relatively bare; it should contain the stars and galaxies visible from the earth, arranged by distance from Earth (or equivalently, time or red-shift). I set out to remedy that.

  • The circular 2D observable universe image, created by the artist Pablo Carlos Budassi, is the perfect complement to the missing interior of the DEUS sphere. It features a logarithmically scaled radial time axis with the earth at the center, surrounded by the solar system, the near Milky Way distorted from the far Milky way, distant galaxies, and, near the periphery, the earliest periods of structure formation. The outer rim corresponds to the big bang, with the cosmic microwave background being the circle slightly interior to it. (Note that this implies that the cross sections should project slightly outside of the CMB sphere, by the "distance" from Big Bang to CMB.) This picture attempts to be astronomically accurate, up to a point. As a 2D image it is one dimension lower than you need to fully capture the sky. The radial dimension is simultaneously time, distance, and red-shift, while the single angular dimension attempts to capture the two angular dimensions of the sky as best it can. It took a lot of Photoshop futzing to divide this into sections and distort them to fit the sides of the cutaway octant in the figure above. Its use of logarithmic scaling allows the solar system and the distant galaxies to be shown in the same image; with linear scaling the Milky way would probably be contained in the central pixel. The same idea could have been used in the timeline to make its cross sections logarithmic, so that the earth-tube could have been shown. The mapping from the timeline to the spherical cross-section would then have been easier to show.
  • A diagram of a hypothetical funnel-shaped universe was adapted to form a light cone linking the left and right images. The idea is that this shows the points in the left (timeline) image that yield the light that is perceived at a moment in the observable universe at right -- the point of the funnel. Everything inside the light-cone would be the past, since its light would already have passed earth by, while everything outside the light cone would be in our future, its light having not yet reached us. Everything on its surface is our present; these are the things we can see now because the light is just arriving now. This may be counterinutitive, since the "past" (funnel interior) is farther away from the sphere, so you may think its light has farther to travel and so may not have arrived yet. However you need to envision the Earth-history world-tube snaking through the middle of the timeline. What matters is not when the light reaches the sphere but when it reaches the Earth-tube, because that is when it would have been (or is being, or will be) seen. The light from the interior of the cone, sloping parallel to the wall, would have intersected the Earth-tube to the left (right in the original image below, which I flipped) of the cone's endpoint, while the light outside the cone, following a similar path, hits Earth in the future. The second funnel-universe image below, which I did not use, has stars emblazoned on its surface. This is a good idea; if  I were better able to render the world-tubes of the timeline diagram, the intersection of those tubes with the light-cone would be the stars and galaxies of the observable universe at right, and it would make perfect graphical sense to highlight them at the cone's surface, desmearing the tubes briefly at those points. (I placed desmeared objects at the end of my tubes, which was easier but doesn't really make sense.) INote also that since the timeline has only two dimensions of space, the time cross-sections of  the cone form a set of increasingly larger circles, which should in principle map to concentric circles in a cross-section of the 3D observable universe such as the one at right. (I thought about adding cross sections to the light cone and corresponding concentric circles inside the sphere, but I decided that would clutter the image too much.) To put this another way, the Budassi 2D image should be draped over the surface of the light cone.                You may be tempted (as I was) to think that the CMB disk at the left of the timeline is somehow wrapped around the observable universe sphere to form its surface, but the situation is actually more complicated than that. The light cone's intersection with the CMB disk of the timeline is a circle; inside that circle is old (past) CMB light that hit Earth a long time ago, while outside is CMB light that will hit us in the future. Since the timeline has one spatial dimension less than it should, that circular intersection is really a sphere, and that is what is on the outside of the observable universe: a spherical shell of CMB light hitting us now. Note that although a light-cone is usually depicted as having straight sides at a 45 degree angle, that would be incorrect here. The 45 degree angle assumes that space and time are in equivalent units, normalized by light speed, but that is not true in the timeline diagram. More importantly, the expansion of the universe implies that the sides of the cone should be curved rather than straight. This is why the observable universe is more than twice 14 billion light years in diameter, even though the universe is only 14 billion years old -- because the cosmic expansion adds to the distance.This particular cone is, unfortunately, not curved the right way -- it should have a nearly constant angle where the universe's size is nearly constant, and should be steeper when the universe is expanding more quickly.  

  • The antique cosmology is from the  Harmonia Macrocosmica of Andreas Cellarius, a star atlas published in 1660. It depicts a geocentric universe (earth at the center), with concentric spheres holding the orbits of the moon, Mercury, Venus, the Sun, Mars, Jupiter, and Saturn. It bears a superficial resemblance to the more modern version of the observable universe depicted to its right. The outermost sphere, Sphaera Zodaici, holds the "fixed stars". I think if Cellarius could see these newer images he would be pleased at the progress that has been made, but he would also feel, with some justification, that he got many of the details right. Although we may smile smugly at his obsolete geocentric universe, long since debunked by Copernicus, the observable universe is in fact geocentric, since that is where we observe from. The central detail of Budassi's (geocentric) diagram is shown below for comparison. 

Postscript 8/15/2016

I have played around with some image distortion in MatLab to try to address some of my critiques of the light cone image above.

First I transformed the Budassi image to polar coordinates (y=angle; x=log dist from earth; Big Bang at right margin).

I then explored various mappings onto the surfaces of cones, initially focusing on curved cones such as the parabolic below, based on my hunch that the cosmic expansion would yield curved light cones:

I also wondered if there would be a point in the past where the expanding light cone would hit the "walls" of the pastward-contracting universe. The image below imposes a sinusoidal envelope on the cone to explore that idea. Physically it turns out to be all wrong but it is rather pretty.

The reason I now think this is wrong is because of the following plot, taken from Ethan's blog, to which I added the light cone line and the yellow horizontal and vertical lines from the intersection of the light cone with the size of the universe. This shows that the intersection would have occurred around 10^4 (10,000) years after the Big Bang. Since Budassi uses a time scale that is logarithmic backwards from the present, this point would be less than a pixel from the Big Bang, whereas my sinusoidal image above has the Milky Way, "only" 200,000 light-years across, mostly in the contracting phase. I considered making a bi-logarithmic time axis, to expand the moments closest to the Big Bang as well as those closest to the present, but that would not solve the problem that the Budassi image contains very little information about those early moments; I would need a different source. (Maybe someone can create a reverse Budassi image with the Big Bang at the center?)


My current best guess is a straightforward light cone shown below. I explored making it transparent and showing the other side with some reduction in intensity, but I couldn't get a satisfactory image, so I opted instead to map all 360 degrees of Budassi onto the visible surface of the cone, so as not to lose any information.

Then there is the question of representing world lines in a timeline. I used a combination of shifting, scaling and truncation to convert the above into the below:

and then added it faintly to the cone:

The divergence is intended to reflect cosmic expansion. World lines of course are not part of the light cone but could reasonably be considered part of its interior (as well as exterior, but I haven't tried to extrapolate beyond the cone yet).

Finally in these explorations I discovered some serious astronomical work along these lines, which was apparently one of the sources that Budassi drew on: a group at Princeton that has been creating accurate geocentric logarithmic maps of the universe such as this:
I tried mapping that onto a light cone, but the results are blurry and less graphically interesting than the Budassi imagery. Sometimes it helps to have some artistry with your science!

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