New big bang model

I had been reasearching aspects of the big bang and was contemplating the acceleration of the expansion of the universe, and wondered how this could best be described

Using the analogy of the north and south poles being the begining and end of time then the shape of the universe through time would best be described as sine wave, or rather the first 180 degrees of the wave

The observations we currently have however seem to match that of a Tan wave and i simply wondered if this was indeed reality.

in a sine wave scenario the pattern of expansion should start out high and gradually decrease over time until it stops expanding and starts contracting ever more rapidly until ending at the big crunch. however current observations suggest the expansion is undergoing accleration rather than slowing down. this is why i thought a sine wave would better describe the expansion.

here hypothetically as the expansion reaches the speed of light this would corespond to that of 180 degrees on the tan wave. rather than the end of the universe occuring as a collapse it would occur in a final super expansion, reaching the speed of light, reverse and begin from the middle again.

This seems more logical to myself, as in the sine wave scenario there is no point at which an infinate density can exist and be accounted for as is desribed in the big bang theory.

The tan wave however seems to accomodate this need. in the exact moment of 180 degrees through the wave what ever the amplitude at 179.9 recouring degrees is, is suddenly switched to the negative of this, but over a pediod of 0 time. this is there fore accounting for the infinate desity at the beggining of time.

A wild idea, yet seems to have at least some merrit. if anybody with physics knowledge could give me a second opinion on this it would be greatly appreciated, as mr hawking is currently quite too busy to speak to me !

Waves:






Edit for graphs,









Final edit, much deliberation on best diagrams