Undergrad Universe Expansion: GR vs Hubble Reconciled

[jeremyfiennes]

TL;DR

The two seem contractory. How are they reconciled?

The GR predictions for the universe's size are those of fig.a. Whereas the Hubble expansion is exponential, fig.b. How are the two reconciled?

 

[PeterDonis]

The GR predictions for the universe's size are those of fig.a.

Where are you getting that figure from?

the Hubble expansion is exponential

What "Hubble expansion" are you talking about? Where are you getting that figure from?

 

[jeremyfiennes]

I don't know. My memory doesn't go back that far. But it us implicit in this one which I found in wikipedia:

"Possible velocity vs. redshift functions; patterned after Davis & Lineweave, CC BY-SA 3.0." The 'linear' curve is the Hubble law case.

 

[Bandersnatch]

All four curves in the OP are allowed under GR. Without knowing the context, we won't be able to answer why they were labelled like that.
The first graph shows the evolution of the scale factor with time in universes with matter+radiation (although the latter doesn't really show in such simplified representation) depending on the ratio of density vs critical density - these are labelled 'flat' and 'closed'. For the one labelled 'open' to curve upwards like that, the composition must include dark energy. If it were approaching linear expansion instead of exponential, all three would represent the possible futures of the universe as could be encountered in textbooks pre-1990s, before dark energy was seriously considered. As is, the 'open' one is a bit out of place in its company, and mislabelled, but permissible under GR nonetheless.
The graph labelled 'b) Hubble' shows the time evolution of the scale factor in universes where the only component present is dark energy. It's the only case where expansion is exponential. It is also equivalent to the Hubble parameter never changing, so it's what you get from naively assuming that the Hubble constant in the Hubble law is constant in time.
All kinds of scale factor evolutions are permissible under GR - exponential, linear, approaching steady state, contracting. Which one is appropriate depends on what's in the universe.

The graph in post #3 shows a completely different thing. That one's about how Doppler shift works.

 

[jeremyfiennes]

Thanks.

 

[PeterDonis]

I don't know. My memory doesn't go back that far.

Need I point out that this means they are not valid references?

Since we have no valid basis for discussion, this thread is closed.