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Figure 3.2 Analogy of the universe as a city laid out in a grid.

Friedmann's first equation is:

8nG A k

3 3 a where H = Hubble parameter, a = scale factor, G = gravitational constant, p = mass density of universe, A = cosmological constant, k = curvature parameter.24 By way of explanation, the scale factor, "a," of the universe is a global multiplier to universe size. Imagine the universe as a perfectly laid out city with streets that travel only north-south and east-west. Streets are spaced at equal-distance intervals. Street intersections then define perfectly symmetric city blocks. One could go further and think of buildings in the city as analogous with galaxies in the universe.

The distance from one city block to another is a function of two values: the originally laid out distance (call that the "normalized" distance) and the scale factor multiplier "a" Note that, as in Figure 3.2, when one multiplies by a scale factor of 1/2, one still has precisely the same city with the same number of city blocks. The only thing that has changed is the distance interval between the city blocks.

Now consider buildings within the city. If the city block distance were reduced to the size of buildings, clearly something must give. The buildings would be squeezed together and destroyed. This is analogous to what happens with matter in the real universe. The sizes of nonelementary particle mass structures such as protons, neutrons, atomic nuclei, and so on are fixed; they do not change with the scale factor. Other physical structures, such as massless particles, do adjust with the scale factor. The wavelength of radiation

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