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johne1618
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To quote cosmologist Ned Wright:
If the Universe is the same at all times, it is argued that the value of the Hubble parameter must be a constant [itex]H_0[/itex], so
[itex] H = \frac{\dot{a}}{a} = H_0[/itex]
leading to an exponential cosmology
[itex]a(t) = e^{H_0 (t - t_0)}[/itex]
where [itex]t_0[/itex] is the present age of the Universe.
This model is an expanding model that does not change its appearance over a time interval from [itex]-\infty[/itex] to [itex]+\infty[/itex].
However this model has been discredited as there is good evidence of a Big Bang Universe with a finite age.
But perhaps one could save the Perfect Cosmological Principle by saying the Universe should look the same at all times in co-moving cordinates.
Now we have
[itex] H = \frac{c}{R_H} [/itex]
where [itex]R_H[/itex] is the Hubble radius.
Therefore if the Universe is to look the same at all times in co-moving co-ordinates then [itex]R_H[/itex] must be co-moving and so
[itex]R_H = a(t) R_0[/itex]
where [itex]R_0[/itex] is constant.
Therefore we have
[itex]\frac{\dot{a}}{a} = \frac{c}{a R_0}[/itex]
As [itex]H_0 = c/R_0[/itex] we have
[itex]\frac{\dot{a}}{a} = \frac{H_0}{a}[/itex]
with solution
[itex] a(t) = H_0 t[/itex]
This model is static in co-moving co-ordinates with a finite lifetime.
The Steady State model of the Universe was proposed in 1948 by Bondi and Gold and by Hoyle.
Bondi and Gold adopted the "Perfect Cosmological Principle", and added the assumption that the Universe was the same at all times to homogeneity (the same in all places) and isotropy (the same in all directions).
If the Universe is the same at all times, it is argued that the value of the Hubble parameter must be a constant [itex]H_0[/itex], so
[itex] H = \frac{\dot{a}}{a} = H_0[/itex]
leading to an exponential cosmology
[itex]a(t) = e^{H_0 (t - t_0)}[/itex]
where [itex]t_0[/itex] is the present age of the Universe.
This model is an expanding model that does not change its appearance over a time interval from [itex]-\infty[/itex] to [itex]+\infty[/itex].
However this model has been discredited as there is good evidence of a Big Bang Universe with a finite age.
But perhaps one could save the Perfect Cosmological Principle by saying the Universe should look the same at all times in co-moving cordinates.
Now we have
[itex] H = \frac{c}{R_H} [/itex]
where [itex]R_H[/itex] is the Hubble radius.
Therefore if the Universe is to look the same at all times in co-moving co-ordinates then [itex]R_H[/itex] must be co-moving and so
[itex]R_H = a(t) R_0[/itex]
where [itex]R_0[/itex] is constant.
Therefore we have
[itex]\frac{\dot{a}}{a} = \frac{c}{a R_0}[/itex]
As [itex]H_0 = c/R_0[/itex] we have
[itex]\frac{\dot{a}}{a} = \frac{H_0}{a}[/itex]
with solution
[itex] a(t) = H_0 t[/itex]
This model is static in co-moving co-ordinates with a finite lifetime.
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