### 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?

pela Correct answer

4 years agoYes, 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.

License under CC-BY-SA with attribution

Content dated before 7/24/2021 11:53 AM

ProfRob 4 years ago

See also https://astronomy.stackexchange.com/questions/10585/evolution-of-the-hubble-parameter https://astronomy.stackexchange.com/questions/11408/hubble-law-cosmological-redshift-and-distance https://astronomy.stackexchange.com/questions/18880/how-is-the-universes-expansion-accelerating-if-the-hubble-constant-is-decreasin?rq=1