Hubble Expansion & GR Universe Size Predictions: Exploring Forces & Results

[jeremyfiennes]

TL;DR Summary

How do they correlate?

Consider a celestial object distance d1 away from an observer, receding at v1=H0d1, Fig.a. And the same object at some time later, now at distance d2 and receding at higher speed v2=H0d2, Fig.b
-- 1) where does the accelerating force come from? 2) the resulting universe-size vs time curvebeing exponential, Fig.c, how does this correlate with the GR predictions, Fig.d?

 

[Ibix]

And the same object at some time later, now at distance d2 and receding at higher speed v2=H0d2

This isn't correct. $H$ is the same at all places in space, but it changes with time. So $H(t_0)=H_0\neq H(t_0+\Delta t)$, which means that at most one of $v_1=H_0d_1$ and $v_2=H_0d_2$ can be correct for a single co-moving galaxy.

1) where does the accelerating force come from?

This is GR. As usual, there is no force, just geometry.

2) the resulting universe-size vs time curvebeing exponential, Fig.c, how does this correlate with the GR predictions, Fig.d?

See my first comment.

 

[jeremyfiennes]

Thanks. Query: we see galaxy A distance d_a away and galaxy B a greater distance d_b away. But the galaxy B is seen further back in time. If H₀ is the same for both, doesn't that imply that it doesn't change with time?

 

[Ibix]

$H_0$ is the current value of the Hubble constant, and is not exactly the correct value to use for any galaxy since light has been traveling for a finite time and $H$ changed over that time. For redshifts less than one the change isn't significant and Hubble's law as you cite it ($v=H_0d$) applies. For large redshifts, though, $H$ has changed and you need to take this into account.

So if you are applying Hubble's law as $v=H_0d$ and expecting this to apply directly to observation then you are assuming that your galaxies are (relatively) near by and hence $H\approx H_0$. At larger distances you need to model how $H$ has changed since the light set out.

 

[Bandersnatch]

Thanks. Query: we see galaxy A distance d_a away and galaxy B a greater distance d_b away. But the galaxy B is seen further back in time. If H₀ is the same for both, doesn't that imply that it doesn't change with time?

$H_0$ is the same for both at the same moment in cosmic time. I.e. we choose a convention for slicing the space-time, and it is at each slice of constant time that the Hubble parameter is the same everywhere.
As we observe the redshifted galaxies - and as you've correctly noted - the further ones are seen as they were at earlier times. At earlier times, the Hubble parameter had higher values (it decreases with time in any universe that is not dominated by variable dark energy, even in a completely empty one). The particulars of how it decreases determine whether the universe is decelerating or accelerating.

The key here is to understand that in regular expansion, without considering dark energy, Hubble's law is not implying acceleration. This is because there is nothing to increase whatever the recession velocity is.
If you imagine an expanding universe without any matter or energy, just empty space, in which we mark two points and describe their relative motion using Hubble's law: V=Hd, it would be the velocity that is constant, while both H and d change. d changes for obvious reasons (things are moving apart); while H describes how long it takes to cover d at velocity V - the longer the distance, the longer it takes.
If you probe the universe at two different places, at equal times, you'll see that Hubble's law holds, and further points are receding proportionally faster. But if you probe the state of the same points at different times, you'll notice that their velocity is constant (again, in an empty universe), while H decreases.

If you were to then start adding components to the universe, they would affect - again - the velocity. Matter and radiation will attempt to retard the recession. Dark energy will attempt to accelerate it.
If you were to take our empty universe, and add only dark energy to it, then - and only then* - the Hubble parameter would be constant in time as well as in space, and the expansion would be indeed exponential.

*as matter and radiation dilute, our universe does approach this state, but never quite reaches it.

 

[jeremyfiennes]

I shall have to think further about this, my starting point having been that H₀ is constant at all places and all times. For the time being: thanks.

 

[Ibix]

I shall have to think further about this, my starting point having been that H₀ is constant at all places and all times. For the time being: thanks.

Can I suggest a critical evaluation of whatever sources you are using might be in order? Even Wikipedia mentions that $H$ varies with time. Also, $H_0$ is the value of $H$ now. It doesn't change because it's defined to be the Hubble constant now, but it isn't physically relevant at any other time. You seem to be using $H$ and $H_0$ interchangeably, which may be confusing you.