Another commonly cited "mental picture" is the raisin loaf analog. Consider bread dough(with a high dosage of yeast) in which are embedded randomly but uniformly spread out numerous raisins. The dough (envision this either in the traditional shape or as a sphere) is then baked in an oven. As it cooks, the entire loaf expands outward indefinitely (in reality this process will eventually stop). With growth, the raisins become progressively farther apart but the individual raisins largely retain their original size. During expansion the raisins farthest from any one chosen as a frame of reference will move the most and hence at a faster clip (velocity) than those close to each other. While helpful in the visualization, this model too has problems. The raisins near or at the edges do not see raisins in all directions, only inward, which violates the Cosmological Principle. And, as one stands back from the baking loaf, the bread "universe" has a conceptual center. So then, this analog is imperfect and aids only in envisioning a part of the expansion picture.
Let's delve into a pictorial way to visualize this notion of expansion with the help of this diagram which presents the process as a 2-D portrayal using circles (the concept depicted works just as well for the 3-D [balloon] version):
From J. Hawley and K. Holcomb, Foundations of Modern Cosmology, © 1998.
The circle on the left depicts a sphere with a radius r1 on which a coordinate system (essentially, latitude and longitude lines) has been traced. That describes expansion from an initial point (radius r0). As expansion continues, the circle on the right now has a radius r2. The coordinate system has correspondingly expanded so that the coordinates of any point, such as locate any of the three "Saturn" discs (note that they remain constant in size even as they separate; this is analogous to the above-mentioned statement that galaxies do not expand in proportion to space expansion), have changed only in scale. From this one can define a basic function called the Scale Factor, given by the symbol R, which describes the changes in dimensions (three-dimensional lengths) in an expanding system as a function of time. This simple equation applies: R(t) = rn/r0, where rn is the radius at some specific time and r0 is the initial radius (for the Universe, the singularity point). Thus, the amount or rate of expansion (or contraction) can be adjusted by a given Scale Factor; if not defining a linear function one value will yield a faster (slower) rate than another that is numerically less (greater). For a given span of time, separations (length spreads) will be greater for higher R's than lower ones. The coordinates are said to be co-moving, that is, they enlarge during expansion but all x, y, and z points referenced to them scale proportionately with R while maintaining their same relative positions. The Scale Factor is a fundamental geometric property that is relevant to a description of an expanding Universe.
Four general modes of change of R with time are depicted in the next diagram. Note that in three of the four cases shown R varies in magnitude with time.
From J. Hawley and K. Holcomb, Foundations of Modern Cosmology, © 1998.
Graph a shows a decreasing rate of expansion, b a uniform or constant (linear) rate, c, an increasing rate, and d a negative rate of expansion (i.e., a contraction).
Note that for a given increase in expansion over some time from t1 to t2, points that are farther apart at t1 expand at progressively greater velocities than those nearer each other; thus, they cover greater distances in a unit of time (we shall see on the next page that this ever increasing velocity outward is associated with progressive increases in wavelengths of light as shown by the redshift phenomenon).
This can be further elucidated with this diagram:
From J. Hawley and K. Holcomb, Foundations of Modern Cosmology, © 1998.
Let the upper row represent the position of three galaxies at t1 and the lower the later expanded location at t2. The elapsed time is (t2 - t1) = Δt (Delta t = a finite interval of time). Initially, each galaxy is separated by a distance d. Following expansion, A is now separated from B by 2d and from C by 4d. Therefore B has moved with respect to A (the observation position) at a recessional velocity (d/t) of (2d - d)/Δt = d/Δt and C from A at (4d - 2d)/Δ t = 2d/Δ t. Thus, the velocity of recession of C with respect to A is twice that of B to A. (Returning to the balloon analogy, one can see that farther dots recede faster than closer ones relative to some dot chosen as the point of observation.) The relative velocities will depend on the Scale Factor. As determined from red shift studies (next page), in this dynamic Universe any two galaxies are moving relative to one another at different recessional velocities which depends on their distance apart; the velocity between one of these galaxies and still a third that is twice as far away will be double (twice) that of the first pair considered.
Some of the recent ideas on the start times for the first stars and galaxies received support and specificity from the WMAP results. The first stars began to form as Supergiants about 200,000,000 million years ago. The first galaxies began to organize some three hundred million (300,000,000) years later. This diagram depicts these stages (from top): 1) initial stages of CBR variations; 2) clots of dark matter prior to organization as stars; 3) the first supergiants; 4) developing galaxies; 5) galaxies after the first billion years.
The time lines for the first stars and galaxies as measured by different space telescopes (JWST is the James Webb Space Telescope planned for 2010; its mission will focus on the early eons of the galaxies, so that the starting time shown above is a "best estimate" for now) are shown in this diagram. Of special import is the new estimate of when the first stars started to form - about 200 million years after the Big Bang.
the pictures help more than all that scientific babble... =/
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