# How is the Universe's Expansion Accelerating if the Hubble Constant is Decreasing?

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?

• This is a great question, but I did want to point out that there are indications that the original research which showed the universe's expansion was accelerating may have been wrong or flawed. See for example this article. Essentially it all comes down to how well we think we understand supernovas (and some people argue we don't really understand them well enough to say the universe's expansion is accelerating). – zephyr Oct 28 '16 at 13:04
• @zephyr - fascinating news, did not know that – Fattie Oct 29 '16 at 11:58
• @zephyr But with little support in the literature - e.g. see arxiv.org/pdf/1702.08244.pdf and arxiv.org/abs/1610.08972 – Rob Jeffries May 12 '17 at 12:24

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.

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.