Does Hubble's Law imply that galaxies are accelerating away

[Elorm Avevor]

Is I understand it Hubble's law states that V = Hd = dx/dt
Solving this differential equation, I got d = kexp(HT) where k is an arbitrary constant.

This implies d²x/dt² = Hv = H²d and dⁿx/dtⁿ = Hⁿd.

However (at least for me), finding the value of k is a problem, for 1, it must vary from galaxy to galaxy because they are all different distances to us. It must also be small because when t tends to 0, x should tend to 0.

My questions are essentially:
Is my maths correct?
Are galaxies moving away exponentially?
Why when approximating the age of the universe do you use 1/H? because this implies galaxies have been moving at a constant velocity since their formation and I don't understand how that's possible from Hubble's law
Thank you

 

[PeterDonis]

Solving this differential equation, I got d = kexp(HT)

This only works if $H$ does not depend on $t$. But in our universe, that's not the case. The idealized case where $H$ is independent of $t$ is called the de Sitter universe:

https://en.wikipedia.org/wiki/De_Sitter_universe

In this universe, there is no matter, just a positive cosmological constant. Our best current model says that our universe will approach this state in the far future. But currently, the density of matter is still large enough that it has a significant effect on the behavior of $H$, making it still depend on $t$. So the solution you found does not describe our current universe.

 

[kimbyd]

Is I understand it Hubble's law states that V = Hd = dx/dt
Solving this differential equation, I got d = kexp(HT) where k is an arbitrary constant.

This implies d²x/dt² = Hv = H²d and dⁿx/dtⁿ = Hⁿd.

However (at least for me), finding the value of k is a problem, for 1, it must vary from galaxy to galaxy because they are all different distances to us. It must also be small because when t tends to 0, x should tend to 0.

My questions are essentially:
Is my maths correct?
Are galaxies moving away exponentially?
Why when approximating the age of the universe do you use 1/H? because this implies galaxies have been moving at a constant velocity since their formation and I don't understand how that's possible from Hubble's law
Thank you

To add a little bit to what PeterDonis stated, I really like this way of examining the situation. It's actually the precise reasoning I use when talking about why a slowly-varying rate of expansion can be described as an accelerated expansion (too often people think that the common phrase "accelerating expansion" means that the rate of expansion is increasing: it is not).

The argument is idealized, however. At present the rate of expansion is still decreasing, but it is doing so slowly enough that far-away galaxies are moving away from us at accelerating rates. Provided the cosmological constant explains our current observations, eventually the rate of expansion will settle to around 80% its current value.

 

[Elorm Avevor]

A quick follow up question.

If Hubble's constant changes over time, surely each distant galaxy you observe would have a different value for H = v/d seeing as their light was emitted at different times in the past so Hubble's constant was different for each of them. But as I understand it, graphs of v against d show a straight line through the origin implying that all observed far away objects have the same value for Hubble's constant irrespective of when the observed light was emitted.

Thanks

 

[PeterDonis]

as I understand it, graphs of v against d show a straight line through the origin

Only for small redshifts. For larger redshifts the graph curves. The specific curvature of the graph is what allows us to conclude that the expansion of our universe was decelerating until a few million years ago, but is now accelerating.

 

[kimbyd]

Only for small redshifts. For larger redshifts the graph curves. The specific curvature of the graph is what allows us to conclude that the expansion of our universe was decelerating until a few million years ago, but is now accelerating.

Much of the time even graphs of longer distance/redshift relationships appear to be nearly straight lines. This is an illusion created by how the graphs are drawn: the axes are not linear, but logarithmic. In a plot where both axes are logarithmic, a curve which follows a power law (e.g. d = z^2) looks like a straight line. The distance/redshift relation isn't exactly a power law, but it's close enough to appear close to straight on such graphs.