If the rate of expansion (galaxy recession velocity) were to fall on the middle part of the upward sweep of the sine wave, then the rate of expansion would appear to be accelerating relative to the predicted uniform rate. If the rate of expansion were to fall on the initial or terminal portions of the upward curve, then the rate of expansion would appear to be decelerating relative to the predicted uniform rate. It therefore is theoretically possible that the volume of space could be expanding exponentially rather than as a cubic function.

If the expansion of the universe were slowing down at just the precise rate, then the rate at which the volume of space was expanding could also be exponential. This would best fit the most elegant of relationships, showing the interdependence of the expansion of the universe with all entropic processes which are also occurring exponentially. It might also allow us to eventually calculate the rate at which the process of universal volume expansion is slowing.

For the rate of universal expansion to be slowing, there must be dark matter within the universe in the appropriate quantity. There is in fact a substantial amount of indirect evidence accumulating which would suggest that there very well might be enough dark matter to cause this to occur.²⁷... If so, the universe would ultimately be shown to be cyclic, undergoing phases of expansion followed by contraction.

These calculations also assume that time is constant, which in actuality may not be the case; and that the universe can be mathematically represented by a sphere, which could also be questioned. A diagram of a sphere expanding at a constant rate shows how the volume doubling time is continuously changing. Each time the sphere doubles its volume it takes a longer time interval to accomplish this feat.

If a graph is constructed, a cubic curve can be demonstrated that is similar to an exponential curve, but with a slope that rises more quickly. On the vertical (Y) axis, each point represents the universal volume each time it doubles; and on the horizontal (X) axis, each point represents the time required for the universe to double in volume.

If this process were reversed, reflecting a universe undergoing contraction at a steady rate, then a cubic curve would be produced similar to an exponential curve, but falling more quickly. Again, either curve could become an exponential curve under the proper conditions if the rate of expansion or contraction were occurring at the prescribed non-linear rate.

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