http://www.people.cornell.edu/pages/jag8/index.html
(The reader may wish to consult the "preface" or "guide" to this paper, which is found at "About the Papers: An Introduction" section V).
Reading cosmology often leaves me wishing I had a simple map of the Cosmos for reference and orientation, just as I want a globe handy when reading the geography or history of Earth. As I have never found a suitable map for this purpose, however, I decided to attempt its production myself, proceeding on the generally held assumption that the Universe has expanded to its present size from very small beginnings. The resulting map is therefore relevant to either "Big Bang" or "Inflationary" cosmology. I found the mapping effort so illuminating and mind-stretching that I feel others interested in cosmology and astronomy generally will find the map and mental exercise it affords both stimulating and useful.
Producing a map of the Universe is not a straightforward problem in the scalar representation of 3-dimensional space. For example, how does one represent the fact that we look backward in time to an ever-smaller Universe as we look outward toward ever-larger regions of space? And how should we indicate the point of origin, or the outer boundary, of our Universe? A three-dimensional model gives us no adequate way to indicate the spatial limits of our Universe because it lacks a crucial dimensional parameter, time.
The mapping problem presented by the time dimension is easily stated: as we look outward in space we look only backward in time. Because of the finite speed of light, we cannot look out in space into the present. We see our Universe not as it is, but as it used to be, in an ordered regression of spatial shells receding into the past as we look deeper into the heavens. Furthermore, the past Universe that we see from Earth is a unique subset of the whole past, as we cannot see any of our own history, and we see only single moments in the history of other parts of the Cosmos. To escape from the observational tyranny of the one-way character of time and the finite velocity of light, we must find a way to disentangle the spatial and temporal dimensions so that we can map what "is" as well as what we are constrained to see. Problems such as these are wholly unfamiliar to the Earth cartographer and require the use of a spacetime map.
At this point we need to recognize that our mapping problem is not so much one of content (the number and positions of galaxies), as of finding an appropriate methodology for representing the universe of spacetime which the galaxies occupy. To state the problem in terms of Earth's geography, cartographers needed to develop spherical models of the Earth before they could make realistic maps of continental positions. The spacetime map I present is analogous to a blank cartographer's globe, containing only lines of longitude and latitude. Although for the present devoid of material content, I think the reader will find there is much to learn even from an empty map of our Universe.
To construct a spacetime map, we must reduce both scale and dimensionality. All 3 spatial dimensions are collapsed into a single line, and time and space are accorded equal status as mapping parameters. The compression of dimensionality results in the loss of recognizable features - the Universe does not "look like" the spacetime map. Nevertheless, the map allows us to represent our Universe in a dimensionally correct manner and at the same time show what we see of it and what we don't. The map helps us to orient ourselves with respect to the spacetime structure of the Cosmos and to understand where we are with respect to the observable and invisible Universe. The deployment of the Hubble Space Telescope, with its exceptionally deep and clear views of space, has increased our need for such understanding.
Figure 1 shows a spacetime map of a Universe 16 billion years old; as of 2003, however, new measurements by the Hubble and other telescopes suggest the Universe is only 13.7 billion years old (Sky and Telescope, May, 2003). For our general purposes, however, the relative size of the map will not affect our discussion or understanding of the model. Where it does matter, we use the new values.
The map consists of 16 concentric, evenly spaced circles which represent the increasing spatial volume of an expanding Universe at billion-year time intervals. The map implies the validity of the modern notion that spacetime is not a void which preexisted the creation event, but is itself a product of the Big Bang. In this representation, the 3 spatial dimensions have been compressed into 1, hence the line of any circle contains all of the space of the Universe at the particular historical moment indicated by its intersection with the time line. The center of the circles represents the Big Bang, or time zero, and the outermost circle represents the spatial volume of the present-day Universe. The circular shape of each space line represents an isometric time curve, that is, all the space in the line is of the same age, as it is equidistant from the Big Bang, the center of the diagram. The shape of the map does not reflect the physical shape of the Universe, but indicates instead its uniform age and finite size.
The time dimension is the radius of the diagram and controls the development of the map. Uniform intervals between spatial circles indicate an even flow of time. Any number of radii could be drawn from the center of the diagram through the spatial circles, like spokes from a hub, to represent the time lines of other observers in galaxies distant from our own. The map represents the Einsteinian connection between time and space in three ways: 1) the time dimension is constructed at right angles to all three spatial dimensions simultaneously; 2) the spatial dimensions are measured in units of light years; 3) time and space are linked by the geometry of the map as the radius and circumference of a circle. Because this latter relationship is linear, the map represents only the change in the cube root of the volume of the Universe per unit of time.
From the viewpoint of any observer, 3/4 of Fig. 1 is imaginary; as I have drawn the map, only the upper left quadrant is real. There are two reasons for this: the asymmetry of the time dimension and mapping artifact. Because time runs in one direction, only the left or right hemisphere of Fig. 1 can be real. Secondly, because the lines of the circles contain all three spatial dimensions, either the upper or lower quadrant of the remaining hemisphere is redundant. We are left with a map more nearly resembling the textbook 2-axis diagrams of spacetime. Nevertheless, I will continue to refer to the full figure because its full symmetry is helpful to generate the real map and because the full figure is more easily visualized.
Figure 1 is appropriate only for a Universe composed entirely of light. The map shows a Universe whose radius in light years is equal to its age, indicating expansion at the maximum possible rate. The spatial circles are evenly spaced and the map is flat, illustrating an historically uniform rate of expansion. If we want our map to represent a Universe containing matter, then we must show a gravitational deceleration of the expansion, that is, a reduced volume increment per unit of time. We can indicate a deceleration by bending the time line out of the plane of the paper, producing a curved map. In a curved map, while the length of any time line increment remains constant, its effective length as the controlling radius of the map is shortened, reducing the rate of growth of the spatial circles.
The greater the gravitational deceleration, the greater the curvature of the map and the more strongly the growth of the spatial circles is suppressed. A change in the degree of curvature represents a change in the rate of deceleration. If the curvature of the map is great enough, it will begin to form a sphere, grow to the region of its "equator", and continuing beyond, start to shrink in size, allowing us to portray a contracting Universe. Time does not "flow backwards" in this representation, its forward motion simply drives a shrinking rather than expanding Universe.
This is simple to visualize on a globe of the Earth. Place the Big Bang origin at the north pole, and the space and time axes become lines of longitude spaced 90 degrees apart. The spacetime arcs correspond to parallels of latitude connecting the two longitude axes. As the longitude axes flow over the globe toward the equator, they become parallel, "pulled together" by the connecting lines of latitude, which have slowed their growth in length as they approach the equator. Flowing past the equator, the longitude axes of space and time converge at the south pole, in the "Big Crunch" of a closed universe. Thus while the mathematics behind all this may be obtuse, visualizing the result is not difficult. Everything we have said about the flat map may be overlaid upon the curved surface.
The actual configuration of our curved map should not be spherical, but egg-shaped, with the poles on the ends of the egg. The egg shape reflects changes in the rate of expansion and contraction, changes which are greatest near the time poles of the Universe.
For a "closed" Universe, which collapses, the full globe shape applies; for an "open" Universe, one which expands forever, the appropriate shape is that of a widely flaring bell, gradually opening to a nearly flat rim. The third possibility is that of a Universe balanced between the open and closed situations, barely changing as time passes. A map of the latter situation would look something like 1/2 of a large egg, continued indefinitely beyond its "equator" as a cylinder. (Recall that, as in the flat map, only one quadrant of the curved maps is real). To facilitate our discussion, I will not refer to any of these curved forms, but only to the flat map of Fig. 1. Anything said about the flat map can be converted to a curved figure by projecting the flat map onto the appropriate 3-dimensional form.
The "shape" of the Universe is a four-dimensional concept, involving time as well as space. This "shape" is the sum total of the "present moment" as it is realized throughout the spatial volume of the Cosmos. If light had an infinite velocity, this is the Universe we would see. This "shape" changes constantly due to the passage of time and the contraction or expansion of the Universe. The outermost circle of the map represents this instantaneous shape, which because of the combined intrinsic motions of time and light is really a "happening" and not a fixed shape at all. At the grand scale of the map, of course, this moving boundary is relatively stable.
From the Earth's position on the outer circle, I have constructed an interior circle which has the time line as its diameter. I call this interior circle the "observer's circle", or "light line". (For the present, the reader should ignore the second interior circle constructed from B). The observer's circle is the path of all light rays between Earth and the Big Bang; all that we can see of our Universe lies on this line (recall that due to the dimensional compression of the map, the light line corresponds to our full 360 degree view of the heavens in ordinary experience).
The observer's circle is one-way, consisting of the paths of light rays received, not sent, by the Earth. Light rays sent from the Earth (or any other source), always move within the (currently) outermost spatial circle, and their paths trace out the observer's circles of the positions which receive them. For example, the paths of light rays sent from Earth 4 billion years ago (arc A'B) to another observer who is now 4 billion light years distant, and the reciprocal exchange (arc B'A), are shown. These light paths result from the combined action of 2 intrinsic motions: the arc A'B, for instance, is the trace of 1) the upward motion (with respect to the map) of the light ray as it moves within its current spaceline from A' toward B, and 2) the radial, outward motion of that spaceline as it ages 4 billion years.
The observer's circle is constructed by drawing tangent lines from the Earth's present position to each of the interior spatial circles, then connecting the points of tangency. The reader may verify, with a straightedge, that these tangent points all lie on a circle which has earth's time line as its diameter. For the sake of clarity, only the tangent lines from Earth and observer "B" to the 12-billion-year spatial circle are included in the diagram.
What is the rationale for our method of constructing the observer's circle? The mapping problem to be solved is to determine the path of a light ray coming from the Big Bang (or any other part of the visible Universe) and seen from Earth today. To do this we must understand how we see our Universe when we look outward into space, then translate this understanding into a mapping procedure.
Any observer of the Universe may be considered to occupy the center of an infinite set of nested observational shells. An observational shell is the 2-dimensional inner surface of a hollow sphere whose radius is determined by the depth of the observer's view into space. The observer sees the Universe as a coherent, 3-dimensional stack of these shells. These shells are 2-D spatial subsets of the Universe as it existed at a particular moment in its history. They are unique to our view. The chief mapping significance of the shells is that they are 2-dimensional surfaces. Because a 3-dimensional volume is represented in our map as a 1-dimensional line, a 2-D surface must be mapped as a zero-D point.
An observational shell of appropriate radius from Earth must intersect each of the interior spatial circles of the map; our task is to find the geometric principle which allows us to identify these points of intersection. Connecting these points from the edge to the center of the map will represent the light path in question. Because the light line must intersect each space line at only a single dimensionless point (from the argument given above), there are just two possible geometric construction procedures which yield solutions to the problem, one of which produces the time line. We cannot see our own past, so the time line is obviously not the solution we seek. The only other possibility is the points of tangency, and it is these which I have used to define the light path of the observer's circle.
Readers should try to generate the following mental pictures to verify for themselves the logic of the preceding discussion: imagine looking out in space to the distance of our Moon. Imagine the complete observational shell that surrounds the Earth at this distance: a 2-dimensional spherical shell of radius (approximately) 240,000 miles, or about 1 1/3 light seconds. This particular observational shell is a spatial slice of the entire Universe as it existed 1 1/3 seconds ago, but the only part of that past Universe we can see is the slice that contains our Moon. Now repeat this imaginary flight to the larger observational shell that contains our Sun, which is a spatial slice of our Universe as it existed about 8 minutes ago (the Sun being 8 light-minutes distant). Although we know the entire Universe existed 8 minutes ago, the only part of it we can see as it existed then is the slice containing our Sun. Finally, imagine the huge observational shell that cuts through our neighboring galaxy, Andromeda, at a distance of about 2.2 million light years. The Andromeda stars intersected by this large observational shell are the only objects we can see in the entire Universe as it existed just 2.2 million years ago. (In all cases I have assumed that the remainder of these shells cuts through empty space). Note that, in principle, we can make the "time thickness" of these shells as thin as we wish, right down to a true 2-D surface.
The exercise above demonstrates that we never see all of our Universe at a single instant of time. Rather, we see successively larger, and always different, portions of it in spacetime shells which recede further into the past as we look deeper into space. Each shell is a 2-dimensional surface which just "touches" the 3-D volume of its historically associated Universe. It should be intuitively clear that these 2-D surface "touches" are simply the higher dimensional analogs of the tangent points on the spatial circles of the map. It is only the combined thickness of an infinity of such shells that gives us the impression of seeing spatial volume. The observational shells become very small as we approach Earth, finally reducing to the size of our own bodies. We see practically nothing of the Universe as it exists "now".
Our "processional" view of the Cosmos is represented on the map in the following way: as the spatial circles recede toward the Big Bang, the observer's circle intersects them at progressively higher positions on their real quadrant arcs. No two spatial circles are seen at the same relative point on their circumferences. The necessity for this arrangement becomes obvious when we recall that each spatial circle contains the whole Universe at a given time; if we are not to see objects in two or more positions at once, we must view these Universes in a sequential spatial as well as temporal progression (contrast the path of the time line in this regard). Gravitational lensing is the exception which proves this rule, dramatically illustrating the structural connection between spacetime and light. The fact that the light line cuts the spatial circles in exactly this special way is one indication that the map's geometry is valid.
Because the time line is 1) the radius of the outermost space line and 2) the diameter of the observer's circle, it follows by simple geometry that, within the map's real quadrant, this spaceline and the observer's circle are equal in length. I interpret this to mean that the Universe we see, even though it is a composite view of many earlier Universes, is of the same spatial volume as the Universe of the present day, which we cannot see. Although the observer's circle was not constructed to represent space, it has space-like properties since it is composed of an infinite series of nested 2-D surfaces. In aggregate, these nested surfaces form the "electromagnetic volume" of the visible Universe.
The map shows a second observer's circle. The second observer (B) is situated 4 billion light years from Earth. We find B's map position, as seen from Earth, by counting back 4 billion years along Earth's time line, then following that spatial circle out to its intersection with Earth's observer's circle; their point of intersection represents our entire observational shell at that distance. We find the second observer's present position in space by constructing B's time line from the Big Bang to its observed position and extending it to the outer spatial circle (this procedure assumes the second observer is not in absolute motion with respect to Earth). When we construct B's observer's circle, we find that it intersects Earth's time line 4 billion years in our past, as it should. This finding is necessary to validate our geometric method. (Note that only the area between the time lines of A and B is a valid map of the simultaneous view of both observers - see appendix.)
The time line of the Earth is another sort of reality. It is the path along which we and other massive objects travel, objects which have intrinsic motion in time, not space. Although we cannot see our own past (except in mirrors), observers elsewhere in the Universe can see some part of it, and influence certainly travels to them and to us from our past. What we may refer to as our own "historical past" is part of the "active past" of some other observer, and vice-versa.
Still a third type of reality is illustrated by the outermost line of the present spatial surface of the Universe. This represents a sort of "universal present moment" that we are continually becoming part of. We receive influence from this sector of the Cosmos only by touch, for this part of the cosmos contains all material objects. We encounter them only in the present moment when we physically touch them. This is the part of the Universe into which we send light signals, rather than receive them. (B's observer's circle is the trace of a light ray sent from Earth 4 billion years ago; it was invisible to us during its entire trip and its arrival "today" in the "universal now" will not be seen by us for another 4 billion years.) Light travels always in the "present moment" of the "universal now". Even though we see the light from the ancient history of distant galaxies, we see that light only in our portion of the "universal now", our personal "present moment".
A fourth type of reality is a composite, the triple intersection of time, space, and light, the reality of the observer's present moment. Our own present is unique in that it is the only type of reality in which we can both generate and receive influence. It is in the present moment that we receive effects from our historical past (timeline: consequences, "karma"), and produce causes for our future. Here we also receive influences from the light universe (information, energy) and send signals into it (communications); finally it is also here that we physically contact and rearrange the material universe in our interactions with solid bodies.
The hemispherical area between our lightline and timeline is an area of spacetime (physically real) which consists of the "stored past". For us, it contains all the history of the Universe that we would or could have seen if we had been watching from the moment of the Big Bang to the present. It is inaccessible to us now, but for others in the Universe, all of it is in someone's light line, or active past, still sending influence and information.
The area shaped like a cornucopia between our light line and the outermost spatial surface is for us a "manifest future", already formed and quite real, of all events we will be able to see and receive influence from that have occurred from the present moment to the Big Bang, but whose light has not yet reached us. Both our "stored past" and "manifest future" are in the real lightline of some observer elsewhere in the outermost spatial surface, the universal "present moment". Simple geometry proves that these two areas are equal in size and must always remain so. The necessary reciprocity between all observers is the intuitive basis for this result.
Another way to state this necessary reciprocity is the realization that if we see B four billion years in his past, then there already exists, for B, a future of equal length (the last four billion years of our history) which will be revealed to him; and vice versa. Hence the light line must split the diagram into equal areas of past and future, and the observer's circle is the shortest line which can accomplish this feat. Since light always travels the shortest path between two points, the circle is the only line which can possibly represent the path of light between Earth and the Big Bang. Note that this proof of the validity of the observer's circle as the true path of light is independent of the others given; it also provides a helpful criterion for the construction of the light line in the more geometrically complex gravitational models.
It is even possible to see our own past; we do so every time we look in a mirror. If we could look into a large mirror in the Andromeda galaxy with a telescope, we would see the Earth as they saw us 2.2 million years ago, that is, 4.4 million years in our past!
The "Inflated Balloon" Model
A popular representation of the Universe is the "expanding surface of a balloon" model. The galaxies are dots painted on the balloon's surface and these dots separate from each other as the balloon is expanded. This is a misleading model, for two reasons: 1) it implies the galaxies are carried along by the expansion of space, which they are not; 2) we are told that we look though the surface of the balloon to see the distant galaxies expanding away from us. This is impossible, because the surface of the balloon represents the present universal moment, which we cannot see at all. The time line of this model can only be correctly represented by extending a line from the interior center of the balloon (its Big Bang origin) to the surface, where it intersects the 2-d representation of space at right angles to all points of the surface membrane. Hence the surface is the universal present, quite impossible to see into, much less through. This is not the surface astronomers have been observing.
This model can, however, be correctly interpreted by slicing the balloon through the center from top to bottom, like slicing a model of Earth through the center from north to south pole. Then superimpose my spacetime map on this section by placing the Big Bang at the core (interior center) of the Earth, with the time line running out to the position of the equator and the space line running up to the north pole. The outermost spatial circle thus becomes the cut surface of the globe running from the equator to the north pole. This is the only way this "inflatable balloon" model can be correctly used, geometrically. The interior concentric circles of the spacetime map are now seen as nested slices of the balloon, separated by equal time intervals during its inflation. The light line is how we visually intersect these past surfaces. The outermost, present surface is accessible to us only by touch.
We live on the edge of the Universe in the present moment of spacetime, as do all other observers. As we look outward in space, we look in every direction backward in time toward the center of our Cosmos, its beginning in the Big Bang. We cannot see beyond the spacetime edge of our Universe into the future, nor beyond its center into a past preceding its origin.
Neither space nor time can be seen in their pure forms, for otherwise we could see the present spatial Universe and our own past. We see only light, an electromagnetic component of spacetime, whose finite velocity constrains our view to a personally unique sequence of spacetime shells receding to the Creation Event. We are fortunate, nevertheless, to be able to see a sample of the developmental history of our Universe, as this information will ultimately be more important for our understanding of the Cosmos than a complete view of its present state.
1) Because of the joint effects of the one-way flow of time and the loss of 2 spatial dimensions, only one quadrant of the full circular map can be real for a specified observer. Furthermore, the map is strictly valid from the viewpoint of only one observer at a time. If interactions between two observers are considered, only the area between them (their shared angle) is a valid map of what both see. The simultaneous view of three observers cannot be represented at all. (I am speaking here of observers widely separated on astronomical scales of distance.) The crux of the problem is that increasing distance from an observer must be represented only in one mapping direction; to do otherwise would recognize positive and negative spatial volumes. When a second observer is placed on the map, the same distance rule must be applied, but the direction of increasing distance for the second observer can only be back toward the first, because the observers must appear in each other's view of the Cosmos. Hence regions of the map which do not lie between the two observers are excluded from mutual mapping interactions (such as the joint sighting of a third external position), because such positions must lie in the "negative space" of one or the other observer.
Although the full symmetry of the spatial circles generates the map conceptually, once we specify the position of the observer, 3/4 of the map immediately becomes unreal. However, the full circular map symmetry is appropriate for a Universe composed only of light, because without the presence of matter it would be quite impossible to choose a specific point of reference on the outer circle from which to orient either time or distance. We can therefore think of the quartered (and curved) map as reflecting the broken symmetry of spacetime occasioned by the presence of matter, and the fully symmetric (and flat) map as representing the primordial generative form.
2) If we construct time-lines from the Big Bang through all the intersections of the Earth's observer's circle and the 15 inner spatial circles, and project these time lines to present-day positions on the outer circle, we find that although these positions are separated by equal increments of time, they are not separated by equal distances on the outer circle. The unequal spacing is a consequence of the decreasing size of the inner spatial circles, and reflects the fact that objects were closer together in the early Universe. Projecting these positions to the outer spatial circle produces a mapping artifact analogous to that causing Greenland to appear inappropriately large on flat maps of the Earth.
3) As we look outward in space, the surface area of our observational shell (the total area of observed sky at a particular distance) increases as the square of the radius of our depth of view. This increase in area is not reflected by the map. Because the observational shell is 2-dimensional, and we have lost 2 of the 3 spatial dimensions in the map, these shells, regardless of size, appear only as the dimensionless points of intersection between the observer's circle and the spatial circles.
4) Some readers may suppose that the arcs of the space lines and observer's circle represent "curved space", or curved paths in space, even in the flat map of Fig. 1. This is not the case. The space lines are curved because they represent space of the same age, and the observer's circle is curved only because the space lines are curved. "Curved" spacetime is caused by gravitation, whose effects are illustrated by the egg- and bell-shaped maps discussed earlier.
5) The "Red Shift". Although not a mapping artifact, it seems convenient to discuss the "red shift" in this section (and see below). There are three types of "red shift"; one is due to the effect of strong gravitational fields (gravitational redshift); a second is due to the effect of recessional velocity (Doppler redshift); and a third is due to the expansion of spacetime, the difference in size between the Universe of source and observer (cosmological redshift). Due to the cosmological redshift, all distant galaxies appear "red shifted" in proportion to their distance: the greater their distance, the smaller the size of the Universe in which we see them as compared to our own, and hence the greater their red shift. From this observation, first attributed to Edwin Hubble and Milton Humason (Mt. Wilson, 1929), we have concluded that the Universe has expanded from very small beginnings to its present immense size.
Consider the confusing observational situation that would arise if we were well along in the collapsing phase of a closed Universe. In this case we would observe both blue and red shifted galaxies, as some earlier stages of the Universe would be larger, of equal size, or smaller than the one we observe from. The spherical form of the map discussed earlier (page 4) will be particularly helpful in understanding such observations. We recall that in the collapsing phase we would be somewhere "south of the equator" on the spherical map; the sizes of the earlier Universes are proportional to the length of the "latitude" lines associated with their time periods. (For example, a Universe at "30 degrees north latitude" would be equal in size to one at "30 degrees south latitude"; in this case, no shift, red or blue, would be seen - if the cause of the red shift is wholly due to the effect of relative size). Universes of larger size than ours can occur on either side of the "equator". The equator's position is determined by the most blue shifted observations (since they come from the largest of all previous Universes); Universes equal to or smaller than ours can occur only in the "northern hemisphere". Hence we can assign our observations to northern or southern hemispheres of the map by correlating color of shift with relative luminosity. It will not be an easy task in any case, but it should be obvious that the map will be a great help in visualizing the problem and making sense of the data.
From this example we discover it is possible (in a collapsing Universe) to see a blue shift which increases with distance from us - as we look "north" toward the "equator" and larger Universes - exactly the reverse of what we observe in an expanding Universe.
6) Finally, I wish to point out the remarkable fact that we are able to construct a four-dimensional map of (flat) spacetime while still adhering to Euclid's rules for geometric constructions, using only a straight-edge and a compass.
A Two-Component Red Shift?
There are three types of "red shift"; see item 5) above. It is easy to see from the Spacetime Map why the cosmological redshift must appear. The spatial circle of the "present moment" (the outermost circle) is the largest in the diagram. On principle, light from any of the earlier, smaller universes (the interior circles) must completely fill the volume of the present universe. Obviously, this will "dilute" the energy and stretch out the wavelength of the light we receive from these earlier and smaller universes, in proportion to their size as compared to ours, producing the cosmological redshift (5, 13). This raises the question: is the observed red shift composed of two contributing factors, one cosmological, from the ratio of sizes, and another Doppler, from recessional velocity?
It seems possible that the map presents us with an opportunity to decide the issue of a one- or two-component red shift. Clearly, the red shift as calculated for the map has only a size differential component, whereas actually observed Z values can have both a size differential and a velocity component. Therefore, if observed red shifts indeed have two components, the Universe will actually be expanding more slowly and hence be older and larger than the red shift data alone would lead us to believe. This is in fact what we are observing now in the supernova data from very deep surveys.
The cosmological redshift would exist if the observed galaxies were not moving at all; it is just a geometric effect involving light and the relative size of light's container. However, it seems the cosmological redshift should enhance a Doppler redshift, much as enlarging a photograph preserves most of its data. Otherwise, information would be destroyed, which is incompatible with the archival function of spacetime. If this interaction occurs, it will lead to a mismatch between red shift and luminosity data, especially at great depth, as we are now observing, since the Universe will actually be older and larger than the exaggerated red shift implies (the true "Hubble Constant" will be lower). This will mean "standard candles" will appear dimmer than expected because they will actually be further away than the red shift alone would indicate; and it will lift the high Z data points on the graph (below) upward toward the more slowly expanding upper gravitational limit line, just as we see.
However, this is not the only possible interpretation of these data. It is quite possible that we see either a Doppler or a cosmological redshift, but never a compounded red shift. The Doppler shift is demonstrated; the cosmological shift is a theoretical assumption. The cosmological redshift also has its origin in velocity, the momentum imparted to particulate matter by the explosive energy of the Big Bang. Precisely because this original velocity is so nearly the same as the velocity of light, the map cannot distinguish between a cosmological red shift due to velocity and one due to size differential - on the scale of the Cosmos or the map, the effect is the same, and calculations from the map performed on the basis of either assumption yield the same result. Therefore, we should keep in mind: 1) interactions between the various red shifts are not well understood; 2) such interactions may (or may not) help explain the recently discovered phenomenon of the "accelerating Universe".
In any case, we want to know how the galaxies moved to their present spatial locations. Were they carried to their current positions by the expansion of spacetime? Or were they blasted there by the momentum their constituent particles acquired from the "Big Bang"? From the ingredients of standard "Big Bang" Theory (12, 13) we can draw some fairly simple conclusions. All galaxies (including our own) move at (nearly) velocity c and are symmetrically distributed throughout space, due to the lengthy period in which their constituent particles were in thermal equilibrium with radiation during the plasma era following the "Big Bang" (lasting approximately 380,000 years). During the plasma era, light and charged particles were coupled, and the particulate constituents of the future galaxies acquired a symmetric spatial distribution, velocity, and momentum that was equivalent to the radiation in which they were immersed. During the plasma era we can say that space, light, and matter all expanded together, and that matter moved and behaved almost as if it were light. At the end of the plasma era ("recombination"), light and space decoupled from matter and went their separate ways. But the symmetric spatial distribution and near-light velocity of matter remain to the present day as a legacy of matter's extended period of thermal equilibrium with radiation during the plasma era.
We see the distant galaxies receding from us at velocities proportional to the divergence of their (constituent particles') initial velocity vectors from our own. The greater the initial divergence in our directions of relative motion, the greater our velocities with respect to each other - then and now. From this, the otherwise curious increase in recessional velocity with distance becomes an obvious result, and does not require a coupling between matter and the expansion of spacetime beyond the plasma era.
This simple model of diverging velocity vectors is reciprocally symmetric between any two observers, but it requires the understanding that the distance parameter (vertical axis ct) behaves like a 4th spatial dimension due to the relativistic velocities of (all) the galaxies. In other words, light-time behaves like a 4th spatial dimension, allowing an extra degree of freedom in where we see the distant galaxies, because their motion at relativistic velocities over cosmological time has carried them to such enormous distances from us. We see them where they were, not where they are, producing an observational 4th spatial dimension.
Another factor producing the observational effect of the symmetric distribution in space of all receding galaxies - an effect that prevents us from discovering our unique orientation or motion in space - is that the Universe can have no visible spatial "edge", because all of space is contained within it. There is no space outside it into which we can look. When we try to look "to the edge of the Universe", which people (including many astronomers), generally seem to think is somewhere in deep space, we can only look backward in time toward the spacetime center and beginning of the Universe (the Big Bang), which completely surrounds us. Since we can only look toward the center of the Universe into more space, we, and all other observers, can only see a symmetric dispersion of galaxies in every direction. Galaxies at greater distance recede more rapidly simply because their original velocity vectors diverged more from our own, so obviously over time their greater relative velocity has carried them further away. But we are all receding from the central spacetime event of the Big Bang with the same velocity in time as well as in space - all observers in the Universe necessarily exist in a region of spacetime on its expanding edge which is exactly as old as the Universe itself, and all see the Universe as we do. The spatial and temporal dimensions of the Cosmos cannot be decoupled from the perspective of any material observer.
The "edge" of the Universe, like its center, is 4-dimensional in spacetime, not 3-dimensional in space. Thus we see the center when we look outward but only backward to the Big Bang, and the "edge" when we try to look forward of our present spacetime position into the future, and discover we cannot - even though much of the future (other than our own) already exists. This is the invisible "void" and the unspecifiable "direction" in which the Andromeda galaxy (for one example) exists "today". Our present position in spacetime is on the "edge" of the 4-dimensional Universe, where we coexist with all other observers in the universal "now", and uniformly move with them into the future. It is the arrow of time, and the very curious fact that the Universe has an actual beginning (apparently correctly intuited by every early indigenous society), that allows us to discover our orientation in the otherwise confusingly symmetric domain of space.
In the velocity vector model, as we look deeper into space, observing the increasing and symmetrically arrayed recessional velocities of the galaxies, we are simply running through the complete inventory, or "Cosmic speedometer", of divergent velocity vectors, from 0 -180 degrees, as they existed relative to our own vector at the moment of "recombination" (the end of the plasma era). Here, greater distance = greater divergence of velocity vectors, while in the map, greater distance = smaller universe. Both are reciprocally symmetric models from the point of view of any two observers, and both may be correct - one does not exclude the other.
The map is an electromagnetic model of an explosive event (the "Big Bang"), a hot mixture of free and bound forms of energy, which distributed the galaxies uniformly throughout spacetime. In the velocity vector model of that event, our "local group" of gravitationally bound galaxies were co-movers (in the form of individual plasma particles) whose initial velocity vectors (at "recombination") diverged not at all, or minimally, from each other; galaxies which we now see halfway to the "Big Bang" diverged from our initial direction of motion by 90 degrees; and the most distant and earliest galaxies we now observe are those whose initial velocity vectors were nearly or quite 180 degrees opposed to our own. As noted above, that this becomes a workable hypothesis only from a viewpoint of 4 "spatial" dimensions rather than 3, is no difficulty considering the size of the Cosmos and the fact that we can only look inward toward its center.
Values for the Red Shift and the Hubble Constant
The red shift parameter (Z) is calculated by the procedure: wavelength observed minus wavelength emitted, this result then divided by the wavelength emitted = Z. In other words, the change in wavelength divided by the original wavelength - the part divided by the whole - equals Z, the red shift, the % change in wavelength. Z can have any value from zero to infinity. The 2.7 degree Kelvin cosmic background radiation is thought to have a Z value of about 1100.
Assuming that the cosmological redshift has its origin in the size difference between the universes of observer and observed, we can directly calculate the red shift we "should" see for each of the billion year intervals of the Spacetime Map, simply substituting the map's radius in years for the wavelength of light. For example, a very distant galaxy observed at the intersection of our lightline with the 4-billion-year spacetime arc (that is, seen when the universe was only 4 billion years old), should have a red shift or Z value of (16-4)/4 = 3.
The "Hubble Constant" (velocity of recession per unit of distance) is the velocity required per unit of distance to collapse (or expand) the universe in the time available. It is calculated here simply by dividing velocity c by the size (radius) of the universe (assuming expansion at velocity c - hence velocity c multiplied by the supposed age of the universe), then multiplying that result by some distance unit of choice, such as a million light years, or a megaparsec (3.26 million light years). The rationale for this procedure is the assumption that all galaxies (in the form of their primordial, constituent particles) are initially blown apart at essentially velocity c, expanding into space almost as if they were particles of light. Gravitational inputs, of course, will somewhat reduce these velocities, "curling" the map, "warping" spacetime. Unfortunately, we do not yet know how much gravity to add to our model - we do not know in what way or by how much the spacetime map should be "bent". However, as these calculated Hubble constant values for expansion at velocity c are almost exactly the same as the recent observational data from the NASA WMAP satellite (Hubble constant approximately 70km/sec/megaparsec for 14 Gyr Cosmos), it appears there is in fact very little gravitational slowing of the universal expansion. (See Sky and Telescope article May 2003 page 16).
In the table below, note that the value of Z rises sharply toward infinity as the universe becomes young and small. I have also calculated recessional velocities directly from the Z values for the 16 billion-year universe. The deeper we look, the more the "ballooning" size differential between the universe of observer and observed, seem likely to complicate the interpretation of red shift data. The problem is surfacing now only because the new generation of large and space-based telescopes can, for the first time, see into the region of exploding Z values. Hence we can expect more observations and reports of "nonlinear phenomena" as the new, large telescopes continue these exceptionally deep observations (see also table footnotes).
Below I list the red shifts or Z values as calculated for universes 12 -18 billion years old, using only the billion-year integers as inputs. In the 16 billion-year universe, recessional velocities have also been calculated from the Z values, and are presented after the Z value in units of thousands of kilometers per second.
For purposes of comparison, in the 15 billion year universe column the second entry is calculated for a closed universe which at 15 billion years is halfway to its maximum expansion (designated "closed 30", or "Zc30"); the third entry in this column is for a closed universe ten times larger than Zc30 (designated "closed 300", or "Zc300"). Note that the size differentials, and hence the Z values, of these closed universes are changing much more slowly than the open case, which has no gravitational input. Note also that it might be difficult to distinguish between the two closed cases observationally, at least in their present assumed stage of development (both only 15 billion years old). We expect observed values for our universe to fall between the open and closed examples. Z values for the "closed 300" universe example will be approximately the same as for a universe which is poised between open and closed (in the first 15 billion years of its development), which is the reason for its inclusion.
The "closed universe" entries were calculated using the relative lengths of the "latitude" lines on a sphere with radius 30 or 300 units (representing 30 or 300 billion light years), beginning from the zero point, Big Bang, or "North Pole", and working "south" 15 units toward the "equator", or region of maximum expansion - see diagram, and the discussion of gravitationally curved 4-dimensional metric surfaces in earlier parts of this paper.
The formula for converting Z values into velocities is (4):
{[(Z+1)sq -1] divided by [(Z+1)sq +1]} times c = res. vel.
(where sq means squared, and c is the velocity of light (300,000 km/sec)
The table assumes no gravitational input (other than "Z30" and "Z300", see above). Gravity exactly equal to the critical density (a universe balanced between open and closed) would reduce the age associated with a given Hubble constant by about 33% (7). At this rate, for example, a Hubble constant of 50 could be associated with a massless universe (such as is represented by the flat map) of about age 19.5 billion years or equivalently, a critical density universe of about age 13.5 billion years. The corresponding figures for a Hubble constant of 60 are about 16.5 and 11.5.
Currently the best estimates for the age of the universe suggest it is 13.7 billion years old, give or take 200 million, and evidence suggests the universe is flat and near critical density, with a Hubble constant around 71 (plus or minus 4) kilometers per second per megaparsec (Sky and Telescope May 2003 page 17); the observed Z data should conform roughly to the scale listed for the 13-14 billion year columns. However, the calculated Z values are maximum values, as they assume uniform expansion at light speed, without any gravitational deceleration (except "Z30" and "Z300", see above). It is obvious that a modest gravitational input is required to bring the higher observational Z values into reasonable agreement with the bottom values of the table (see the table footnotes and column "Z300" values). (First and last values in the table columns are interpolations; the last assumes a Z value for the Big Bang or cosmic microwave background radiation of 1000.)
While the approximations of the table and the data inputs are very "rough", they do not suggest a doubling of the red shift. However, the table values are good enough to confirm that the map "works" empirically and operationally (see graph) - it is properly constructed in that observational data can be sensibly plotted from it, and we can fairly assume that it would work even better if we knew how to "bend" it gravitationally. Note in this regard that the value of the Hubble constant calculated for our 14 Gyr map Universe (69.8) is almost exactly the same as the observed value (71) obtained (at considerably greater cost) by the recent NASA WMAP satellite (Sky and Telescope May 2003 page 17).
ASSUMED AGE OF UNIVERSE (top of table is present moment; bottom is Big Bang origin) |
18 Billion Year Age Hub. Const. = 54.3 km/sec/Mpc |
17 Billion Year Age Hub. Const. = 57.5 km/sec/Mpc |
16 Billion Year Age Hub. Const. = 61.1 km/sec/Mpc |
15 Billion Year Age Hub. Const. = 65.2 km/sec/Mpc |
14 Billion Year Age Hub. Const. = 69.8 km/sec/Mpc |
13 Billion Year Age Hub. Const. = 75.2 km/sec/Mpc |
12 Billion Year Age Hub. Const. = 81.5 km/sec/Mpc
Age | Bil Yrs
Z
|
Age | Bil Yrs
Z
|
Age | Bil Yrs
Z=red shift; | R=recess. Velocity (1000 km/sec) Z; R
Age | Bil Yrs
Z open; | Z closed 30; Z closed 300 Zo; Zc30; Zc300
Age | Bil Yrs
Z
|
Age | Bil Yrs
Z
|
Age | Bil Yrs
Z
|
18
|
0.03
|
17
|
0.03
|
16
|
0.03; 9.5
|
15
|
0.04; 0.01; 0.01
|
14
|
0.04
| | 13 | 0.04 | 12 | 0.05 |
17 | 0.06 | 16 | 0.06 | 15 | 0.07; 19 | 14 | 0.07; 0.02; 0.03 | 13 | 0.08 | 12 | .08 | 11 | .09 |
16 | 0.13 | 15 | 0.13 | 14 | 0.14; 39 | 13 | 0.15; 0.05; 0.07 | 12 | 0.17 | 11 | .18 | 10 | .2 |
15 | 0.2 | 14 | 0.21 | 13 | 0.23; 61 | 12 | 0.25; 0.08; 0.12 | 11 | 0.27 | 10 | .3 | 9 | .33 |
14 | 0.29 | 13 | 0.31 | 12 | 0.33; 83 | 11 | 0.36; 0.12; 0.16 | 10 | 0.40 | 9 | .44 | 8 | .5 |
13 | 0.38 | 12 | 0.42 | 11 | 0.45; 107 | 10 | 0.50; 0.16; 0.22 | 9 | 0.56 | 8 | .625 | 7 | .71 |
12 | 0.5 | 11 | 0.55 | 10 | 0.6; 131 | 9 | 0.67; 0.21; 0.28 | 8 | 0.75 | 7 | .86 | 6 | 1 |
11 | 0.64 | 10 | 0.7 | 9 | 0.78; 156 | 8 | 0.88; 0.27; 0.36 | 7 | 1.0 | 6 | 1.16 | 5 | 1.4 |
10 | 0.8 | 9 | 0.89 | 8 | 1.0; 180 | 7 | 1.14; 0.35; 0.45 | 6 | 1.33 | 5 | 1.6 | 4 | 2 |
9 | 1.0 | 8 | 1.13 | 7 | 1.29; 204 | 6 | 1.50; 0.44; 0.57 | 5 | 1.8 | 4 | 2.25 | 3 | 3 |
8 | 1.25 | 7 | 1.43 | 6 | 1.67; 226 | 5 | 2.0; 0.57; 0.72 | 4 | 2.50 | 3 | 3.33 | 2 | 5 |
7 | 1.6 | 6 | 1.8 | 5 | 2.2; 247 | 4 | 2.75; 0.73; 0.92 | 3 | 3.67 | 2 | 5.5 | 1 | 11 |
6 | 2.0 | 5 | 2.4 | 4 | 3.0; 265 | 3 | 4.0; 0.98; 1.22 | 2 | 6.0 | 1 | 12 | 0 | 500 |
5 | 2.6 | 4 | 3.25 | 3 | 4.3; 279 | 2 | 6.5; 1.41; 1.71 | 1 | 13 | 0 | 500 | . | . |
4 | 3.5 | 3 | 4.7 | 2 | 7.0; 291 | 1 | 14; 2.38; 2.82 | 0 | 500 | . | . | . | . |
3 | 5.0 | 2 | 7.5 | 1 | 15.0; 298 | 0 | 500 | . | . | . | . | . | . . |
2 | 8.0 | 1 | 16.0 | 0 | 500; 299 | . | . | . | . | . | . | . | . |
1 | 17.0 | 0 | 500 | . | . | . | . | . | . | . | . | . | . |
0 | 500 | . | . | . | . | . | . | . | . | . | . | . | . |
"BIG BANG" or "CREATION EVENT": ORIGIN OF COSMOS, TIME BEGINS | |||||||||||||
John A. Gowan 6 May 2001 Reproduction Permitted with Attribution http://www.people.cornell.edu/pages/jag8 |
Z Estimates From Various Sources - for Comparison with Table Calculated Entries (Gyr = billion years); (Note that 1+Z = factor of expansion); (See also graph of 13.5 Gyr Cosmos).
1a) Z = 0.1685 = 2 Gyr distant (Sky and Telescope July 2003 page 18).
2a) Z = 0.25 = 3Gyr distant (Sky and Telescope Dec. 2003 page 19).
3a) Z = 0.3 = 3.5 Gyr distant; Science 293(5533):1273-8 17 Aug. 2001;
4a) Z = 0.5 = 5 Gyr distant; Science 293(5533):1273-8 17 Aug. 2001.
5a) Z = 0.6 = 6 Gyr distant; Science 293(5533):1273-8 17 Aug. 2001.
6a) Z = 0.8 = 7 Gyr distant; Science 11 April 2003 page 270 - 4;
7a) Z = 1.0 = 8 Gyr distant. Science 293(5533):1273-8 17 Aug. 2001;
8a) Z = 2.15 = 11 Gyr distance (fits in 16 bil yr Cosmos; fits younger Cosmos with gravity); Sky and Telescope 103(6):19.
9a) Z = 3 = 2 Gyr old universe (fits at about 3 Gyr old) (Science 23 Jan 1998 page 479);
10a) Z = 4.0 = 12 Gyr distant or 11 % of present age (fits 15 bil Cosmos) (Sky and Telescope 102(3):44).
11a) Z = 5.0 = 1Gyr after Big Bang (Nature Vol. 421 page 329 Jan. 23, 2003);
12a) Z = 6 = 13 Gyr ago; Data from Chandra X-ray Observatory - Sky and Telescope Aug. 2002;
13a) Z = 10 = few hundred million years after BB. Science 293(5533):1273-8 17 Aug. 2001.
14) Distant supernovae are unexpectedly dim, hence more distant (brightness/red shift); expansion rate is 15% greater now than when universe was 1/2 its present age (data from 7 billion light year distant supernovae) (supernova red shifts in these observations range from Z = 0.18 - 0.83) (Science 18 Dec 1998 page 2156).
15) Because the universe is constantly converting its original mass into light (via nuclear fusion/fission and gravity, especially quasars), but no known process adds to the original mass, we expect the total gravitational field of the universe to decrease with time (since light (free energy) produces no gravitational field). Hence a small "acceleration" of the Cosmic expansion (actually a small reduction in the rate of deceleration) is to be expected from this mass/gravity loss. However, If the early Universe converted mass to light at a much higher rate than today (vigorous star formation, galaxy mergers, quasar and black hole formation), a significant reduction to the total gravitational field during that era could result.
It has been objected that the conversion of bound to free energy in stars is not sufficient to account for the observed deceleration of the Cosmos; however, if the conversion of bound to free energy by any mechanism whatsoever occurs in the "dark matter" or "dark energy" presumed to comprise 95% of the substance of the Cosmos, then such universal processes, driven by the conservation of the symmetry of free energy, whether that energy is "dark" or "light" (as demanded by "Noether's theorem"), might well be sufficient to account for the observed lessening of the gravitational deceleration. Finally, the existence and decay of a fourth, heavy "leptoquark" neutrino might even account for the deceleration within the parameters of "ordinary" baryonic matter. (see: Entropy, Gravitation, and Thermodynamics).
16) The graph of the 13.5 Gyr Cosmos (Fig. 2 below) demonstrates empirically that the map is correctly constructed - observations reported in the literature (above) fall on or between "limit lines" calculated directly from the map - as they should. As for the "accelerating Universe", a small effect of this sort is to be expected in any case (see above). The data as graphed suggest the Universe will expand forever.
I have included two extra graphs of the same data plotted for a 12 and a 14 Gyr Universe. Note that the position of the data line with respect to the critical gravity and the no-gravity lines is quite sensitive to our estimate of the age of the Universe. At 12 Gyr we see a strongly accelerating Universe; at 14 Gyr we see no acceleration at all. This makes sense in that if we try to stuff this fixed data line into a too-small Universe, a gravitational effect must appear somewhere to slow down the expansion, and naturally it appears near the Big Bang. The sensitive part of the graph is of course the high-Z end of the zero-gravity line, which balloons upward as the Universe shrinks; the low-Z end hardly changes at all. This reinforces the point made earlier that observations made at great depth are not only especially difficult technically, but especially liable to errors of interpretation due to the exaggerated red shift at these great distances.
Leaving aside these difficulties, perhaps the simplest explanation for the "accelerating Universe" is twofold; 1) as the Universe ages, there is in fact less total gravitational force due to the conversion of bound to free energy in stars, quasars, and black holes; 2) due to Newton's inverse square law, the Universe responds to its own self-gravity more strongly when it is small and young than when it is large and old.
3b) Z = 0.36 = 3+ Gyr distant (good fit at 4 bil in 15 bil Cosmos) (Sky and Telescope 102(3):35).
6b) Z greater than 0.8 is halfway to Big Bang; Science 293(5533):1273-8 17 Aug. 2001.
7b) Z = 1.7 = 10 Gyr distant (supernova); Sky and Telescope July 2001 page 20; (fits 16 bil yr Cosmos, or younger with gravity).
9b) Z = 3.36 = 2 Gyr old Cosmos; Science 7 Nov. 2003 page 951-2.
11b) Z = 5.58 = 4% of age of Cosmos (about 0.56 Gyr; requires gravity to fit anywhere); Sky and Telescope 103(1):17;
11c) Z = 5.8 = 13 Gyr distant quasar in 14 billion year old Cosmos (fits at about 2 bil years old; better fit with gravity) (Sloan Digital Sky Survey).
12b) Z = 6.28 - Quasar; Cosmos less than 1/7.3 present age and 800 million years old (fits at about 2 bil years old; better fit with gravity) (Sloan Digital Sky Survey: Nature 27 June 2002 page 905);
12c) Z = 6.65 = 0.780 Gyr after Big Bang; (requires gravity to fit anywhere); Sky and Telescope 103(6):28.
Note to readers: This postscript has been added to the original paper in response to a recent Scientific American article in which an inappropriate illustration of spacetime was used to demonstrate the "cosmological horizon problem". While it does directly point up the general need for a better spacetime map, it is not necessary to a reading of my original paper. The postscript will be most useful to those with the Sci. Am. illustration in hand (unfortunately, not available from their website.)
The "cosmological horizon problem" consists of the notion that widely separated regions of our Universe cannot have had enough time, given the finite velocity of light, to communicate with each other since their common origin in the Big Bang. Although "their common origin" makes this notion seem ludicrous on first encounter, it has an observational basis, as illustrated in a recent Scientific American article (January 1999, vol. 280 no. 1, page 69). In this example, two galaxies, each 12 billion light years distant from Earth, but seen in opposite directions, cannot have had enough time since the beginning of the Universe to "see" each other, since the Universe is less than 24 billion years old. This supposition, however, is at odds with data regarding the cosmological background radiation, which is uniform to within one part in 100,000. If widely separated regions of the Universe have never been in communication with each other, how could the great uniformity of the background radiation, which implies a period of general cosmic and thermal equilibrium, have been established? The "inflation" theories of Alan Guth and others have been proposed to resolve this paradox.
Considerations stemming from a properly constructed four-dimensional "Spacetime Map of the Universe" (see below), however, suggest the "horizon" problem is bogus, resulting from a misinterpretation of how we see the Cosmos generally and at great depths particularly.
The Sci. Am. illustration is a typical example of this common misperception. Here, the two galaxies are shown at the edge of the Universe, while observers on Earth are shown at the center. This is exactly the reverse of the actual situation; as the spacetime map makes clear, we view the Universe from its edge, looking backward in time toward its center. We can only look backward in time as we look outward in space, to a center which surrounds us, the background thermal radiation of the Big Bang itself. The Universe has a center in spacetime, not in space.
When we construct a geometric four-dimensional spacetime map, it at once becomes obvious that our common perception that faraway objects lie at the edge of the Universe is completely wrong: we are at the edge, since the present moment is the furthest in time and space from the Big Bang. The observed galaxies of the Sci. Am. illustration should be placed toward the center of the Universe, sunk in the past, much closer than we in time and space to the origin of the Cosmos. Placing the observed galaxies at the edge of the Universe rather than the center of the illustration is the reason why the expansion of the Universe appears to cause the "horizon" problem, whereas in fact the expansion solves the problem. Of course, it is impossible with the type of diagram utilized by Sci. Am. to portray the actual four-dimensional situation.
The relevant question is not simply how far these galaxies are from us (which we can directly measure), but how far they are from each other (which we can only infer), in the much smaller Universe they existed in 12 billion years ago, the era in which we now see them. The fact that these galaxies are distant from us does not mean they are distant from each other; in actuality, given the same angle of observed separation (in this case, 180 degrees), the further they are from us, the closer they will be to each other, since more distant Universes are smaller.
Presumably, we know these galaxies are 12 billion light years distant because of their comparable "red shifts". The red shift is telling us that the spatial radius of the Universe from which their light comes is significantly smaller than our own. On principle, the light from their smaller, early Universe must nevertheless uniformly fill the larger radius of our present day Cosmos, stretching its wavelength, which produces the "red shift". Hence these galaxies cannot possibly be separated from each other by 24 billion light years, since the Universe in which both reside is only about three billion light years in diameter (if our current Universe is fifteen). The perceptual paradox is that the further out into space we look, the wider becomes the radius of our observational sphere, but the smaller becomes the Universe we observe.
However, if we know the red shift of the galaxies in question, then we can determine their maximum spatial separation. The only problem with the red shift is that it simply gives us a ratio between sizes, not the sizes themselves. We must know by some independent means the size of either our Universe or that of the observed, before we can translate the ratio to actual distances. Here we use the estimated age of the Universe (15 billion years) and assume it has expanded uniformly at velocity C, without taking gravitational deceleration into account. The red shift of these galaxies would have to indicate that our present Universe is 5 times larger than theirs, for this particular choice of numbers (and assumed conditions) to agree.
Another misrepresentation in the illustration which contributes significantly to the illusion of a horizon problem is that the Earth is shown (in the top panel) receiving light from the galaxies as if all three were in the same present moment of our time. Clearly this is impossible, and directly contradicts the stated conditions that these galaxies are 12 billion light years from Earth. This explicitly means that we see them where they were 12 billion years ago, not where they are "now" (our time). Where they are now will not be revealed to us until 12 billion years in our future. We see them "now" in their 3 billion-year-old Universe (if ours is fifteen); hence as we see them, their maximum separation from each other is 3 billion light years, not 24.
Finally, the notion that we are just now receiving light from these galaxies is spurious, another casualty of the misrepresented time scale of the top panel. A billion years ago we would have received light from their evolutionary predecessors; a billion years before that, from their elemental constituents; and a billion years before that, they would have merged into the Big Bang and our cosmic background radiation (thereby raising its temperature), where their energies would be thoroughly mixed with the Cosmos of that time.
Four Thoughts About the Map and the "Horizon" Problem
A. The question of the maximum separation of two objects in spacetime - is it related to the radius, diameter, or circumference of the Universe - is something of a brain-teaser. The radius of the Earth is approximately 4,000 miles, its diameter 8,000, its circumference 24,000, and the maximum separation of objects upon its surface 12,000. None of this math works in 4-dimensional spacetime. The question is most easily resolved by considering our own situation. Assuming our Universe is 14 billion years old, and has expanded at the speed of light, we are currently 14 billion light years from the Big Bang, the origin and center of spacetime. This is the maximum separation between any two objects in our Universe - nothing can possibly be further from us than the Big Bang and still lie within our Universe. Therefore, the length of the time line determines the maximum separation between objects in any of the smaller Universes we observe - if they have a time line of 3 billion years, then 3 billion light years is the maximum separation between any two objects in that Universe. Hence there is always enough time in any Universe for all its constituent parts to reassemble at their point of common origin, if they move at velocity C.
B. Since the size of our observational shells increases with distance, one might suppose the observed density of galaxies would decrease with depth of view. However, the increasingly large observational shells intersect increasingly densely populated regions of space, since the smaller universes nevertheless contain the same total number (approximately) of galaxies as the larger ones. The two effects counterbalance each other, so that the density of galaxies does not appear to diminish with distance - as observed. There will, of course, be some evolutionary effects, including a sudden cutoff beyond the era of galaxy formation.
C. The very early Universe expanded at less than C since light (the driving force of the expansion) was in thermal equilibrium with matter. This is the period of mixing. Expansion at C occurred only after "recombination", about 380,000 years after the Creation Event. The uniformity of the background radiation, and of the distribution and velocity of galaxies, is the consequence of this long period of thermal equilibrium, which also resulted in the equipartition of momentum among all the massive particles.
D. In effect, due to the finite velocity of light, we "peel" the electromagnetic images of the galaxies away from their spatial positions on the outer circle of the Map. Beginning at the triple point of contact between light, space, and time in the observer's present moment (Earth's position), as we proceed into the past the light line falls progressively further and further away from the actual spatial position of the galaxies (the outer circle), until at the opposite pole of the Universe, light, space, and time meet again in the Creation Event.