I am just getting into the field of cosmology and was wondering what it means practically to say that the Hubble Constant is decreasing while the expansion rate of the Universe is accelerating. I am not sure how this phenomenon is possible. How do we observe this?
It is my understanding that the Hubble Constant is the rate at which the Universe is expanding, and it is decreasing with time. (Although it is asymptotic)... Is this conception flawed?
The Hubble parameter is defined as the rate of change of the distance between two points in the universe, divided by the distance between those two points. The Hubble parameter is getting smaller because the denominator is getting bigger more quickly than the numerator.
In the future, the cosmological constant, $\Lambda$ could result in an exponential expansion with time. A simple piece of maths shows you that the Hubble parameter is a Hubble constant only for an exponential expansion.
For example, suppose $H(t) = (da(t)/dt)/a(t)$, where $a(t)$ is the distance between two arbitrary points in the universe at epoch $t$. Now let's have an expansion rate $da/dt \propto t$ (ie accelerating). But in this case $a(t) \propto t^2$ and $H(t) \propto t ^{-1}$ (ie decreasing with time).
Some extra details:
The solution to the Friedmann equation in a flat universe is $$H^2 = \frac{8\pi G}{3}\rho + \frac{\Lambda}{3},$$ where $\rho$ is the matter density (including dark matter) and $\Lambda$ is the cosmological constant.
As the universe expands, $\rho$ of course decreases as $a(t)^{-3}$, but $\Lambda$ remains constant. Thus the first term on the RHS becomes unimportant if indeed $\Lambda$ is a cosmological constant.
Thus the Hubble "constant" actually decreases from its current value $H_0$ and asymptotically tends towards $ H = \sqrt{\Lambda/3}$ as time tends towards infinity.
All of these answers start with the formula and say therefore $H_0$ decreases, but I think I have a more intuitive explanation:
If you imagine the universe as an open straight rubber band, and the expansion of the universe stretching markings (galaxies) on the band, and say one end is being held stationary (this is us), then the rate at which each galaxy moves away from us is proportional to two things: how fast the rubber band is stretched, and the fraction of its distance over the entire length of the rubber band - why more distant galaxies move away faster.
Then you can imagine by the time a galaxy gets to be the same distance away as another galaxy was before, the expansion of the universe could be accelerating and the band is increasing in length faster, but now the distance to the galaxy is a much smaller fraction of the total length of the rubber band - and hence, depending on the function of time giving the length of the string, these two factors can multiply to give the speed of a galaxy at a certain distance to be decreasing.
Hence as v=$H_0$d, $H_0$ decreases.
It really comes down to what you define as accelerated expansion.
Usually accelerated expansion is taken to mean that the first deriviative of the scale factor $a'(t)$ is increasing. However the Hubble parameter is given by:
$$H \equiv \frac{a'(t)}{a(t)}$$
As in expanding Universe $a(t)$ is increasing, it is possible for $a'(t)$ to be increasing, but $H$ to be decreasing.
I suppose it could be argued that an increasing $H$ (i.e. some kind of phantom energy scenario) is what we should truly reserve the term "accelerated expansion for. E.g. you could argue that a de Sitter Universe where $H$ is a constant, and therefore also any Universe where $H$ is decreasing, shouldn't be described as as "accelerated expansion", as there is effectively no difference between the expansion at some time $t$ and some later time. However we don't go with this definition.
The Hubble Law is just a approximation for "small" scales. When you look for a large scales the scale factor is like
$$a(t) = a(t_0)[1+H_0(t-t_0) + ...+]$$
So, you have another contributions to the scale factor, which affects the value of the hubble's parameter.
So, this parameter is not a angular coefficient of a line. If it would were so the constant need to be increase in a accelerate universe. As this law is just a approximation, then this is not necessary.
$H_0$ and you will get $H_0$.
– peterh
Mar 19 '20 at 18:10
So The Hubble "Constant" has been and still is getting smaller since the beginning of time. It's at about 70 km/s/Mpc and will stop at about 60 or "H=Λ/3". Its getting smaller at a slower rate originally because the matter density of the universe is decreasing more slowly but now the rate-of-H-slowing has decreased even more because the Universe is feeling the acceleration as Dark Energy dominates. "Acceleration of the expansion today just means that the Hubble parameter is decreasing less rapidly than before. But it's still decreasing." Eventually the acceleration will reach a peak and H will be a constant. That is paraphrased from the physics stack, so I don't want credit, which has a much clearer explanation than here above but details above still educate.
Source https://www.physicsforums.com/threads/why-is-the-hubble-sphere-expanding.616551/
Note that if all velocities remained constant, then as distances increased, the expansion RATE, $H_0$, would decrease because it is defined as velocity divided by distance. If $H_0$ is constant over time, then as each galaxy's distance increases, its velocity increases. So, while the expansion rate is asymptotically decreasing to a constant, the galaxies are accelerating. The expansion rate is not increasing, the expansion is.
