### Does the Hubble constant depend on redshift?

• I know there are lot of questions on the Hubble constant already, but I am curious to know if it changes with redshift? If at current redshift, $$z=0$$, we know its value to be 0.7, will it be different at higher redshift ($$z=0.1$$)? If so, is there any relationship with redshift?

4 years ago

Yes, definitely.

The Hubble constant describes the expansion rate of the Universe, and the expansion may, in turn, may be decelerated by "regular" matter/energy, and accelerated by dark energy.

It's more or less the norm to use the term Hubble constant $H_0$ for the value today, and Hubble parameter $H(t)$ or $H(a)$ for the value at a time $t$ or, equivalently, a scale factor $a = 1/(1+z)$, where $z$ is the redshift.

The value is given by the Friedmann equation:
$$\frac{H^2(a)}{H_0^2} = \frac{\Omega_\mathrm{r}}{a^4} + \frac{\Omega_\mathrm{M}}{a^3} + \frac{\Omega_k}{a^2} + \Omega_\Lambda,$$
where
$\{ \Omega_\mathrm{r}, \Omega_\mathrm{M}, \Omega_k, \Omega_\Lambda \} \simeq \{ 10^{-3},0.3,0,0.7 \}$
are the fractional energy densities in radiation, matter, curvature, and dark energy, respectively.

For instance, you can solve the above equation at $z=0.1$ and find that the expansion rate was 5% higher than today.

Since everything but dark energy dilutes with increasing $a$, $H(a)$ will asymptotically converge to a value $H_0\sqrt{\Omega_\Lambda} \simeq 56\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$.

The figure below shows the evolution of the Hubble parameter with time:

As noted by KenG, the fact that $H$ decreases with time may seem at odds with the accelerated expansion of the Universe. But $H$ describes how fast a point in space at a given distance recedes. Later, that point will be farther away, and so will recede faster. From the definition of the Hubble parameter, $H\equiv\dot{a}/a$, multiplying by the scale factor shows the acceleration $da/dt$:

And just to stave off any possible confusion surrounding that wonderful answer, when people talk about the expansion "accelerating," they are talking about what is happening to the expansion speed H times a, not the expansion rate H itself. So your result shows that while H is dropping with or without dark energy, dark energy makes H times a rise with a, whereas matter alone makes H times a drop with a.

@KenG Yes, that's an important point.

If an animal grew by 1% per year, a gedanken microbe on its skin might claim that the expansion was accelerating!

@JohnDuffield Gedanken microbe :D

Good to see this one is finally settled.