The Hubble Constant and Natural Unit

[micomaco86572]

Could we set the natural value Hubble constant to be 1 in some calculation of cosmology, like what we do in the natural unit?

 

[Chalnoth]

Could we set the natural value Hubble constant to be 1 in some calculation of cosmology, like what we do in the natural unit?

As there isn't anything fundamental about the Hubble constant, this doesn't make sense to me. The Hubble constant is just the current average expansion rate. It's different now than it was a few billion years ago, and from what it will be in a few billion years.

 

[Wallace]

Could we set the natural value Hubble constant to be 1 in some calculation of cosmology, like what we do in the natural unit?

Effectively yes, there are many quantities dealt with in Cosmology in which the Hubble constant value is part of the unit, for instance you can say the distance to some object is X Mpc/h where h = Hubbles constant / 100. The /100 part is just to make little h of order unity, which is what cosmologist tend to like their parameters to be.

There are many more examples of this, since so many things just have the actual value of the Hubbles constant as a simple factor.

 

[Chalnoth]

Effectively yes, there are many quantities dealt with in Cosmology in which the Hubble constant value is part of the unit, for instance you can say the distance to some object is X Mpc/h where h = Hubbles constant / 100. The /100 part is just to make little h of order unity, which is what cosmologist tend to like their parameters to be.

There are many more examples of this, since so many things just have the actual value of the Hubbles constant as a simple factor.

Somewhat pedantic note: h = H_0 / (100 km/s/Mpc). It's sort of a way of saying, "Well, this quantity I'm measuring depends upon the actual value of H_0, but we don't know what the actual value is, so we'll just calculate everything based on H_0 = 100km/s/Mpc and carry over the difference between this and the true value by keeping track of the appearances of 'h'."

So I guess this is sort of similar, in a way.

 

[Wallace]

Agreed. The point is that for the most part in cosmological calculations (distance measures dependence on cosmology for instance), the actual value of H0 is simply a normalisation constant and hence can be considered in the above way. This is not true for most other parameters. You couldn't do this with say $$\Omega_m$$.

 

[Chalnoth]

Agreed. The point is that for the most part in cosmological calculations (distance measures dependence on cosmology for instance), the actual value of H0 is simply a normalisation constant and hence can be considered in the above way. This is not true for most other parameters. You couldn't do this with say $$\Omega_m$$.

Well, that depends upon whether your measurement is sensitive to the density fraction or the total density. The CMB, for instance, is sensitive to the total density of normal and dark matter, while supernova measurements are only sensitive to the density fraction. So for CMB measurements, an estimate of $$\Omega_m$$ would indeed depend upon $$h$$, which is why for CMB experiments constraints are usually quoted on $$\omega_m$$, where $$\omega_m = \Omega_m h^2$$.

 

[micomaco86572]

Agreed. The point is that for the most part in cosmological calculations (distance measures dependence on cosmology for instance), the actual value of H0 is simply a normalisation constant and hence can be considered in the above way. This is not true for most other parameters. You couldn't do this with say $$\Omega_m$$.

I think the reason why we cannot set $$\Omega_{m}$$ to be 1 is dimensionless.

 

[Chalnoth]

I think the reason why we cannot set $$\Omega_{m}$$ to be 1 is dimensionless.

Ah, after reading this, I realize that I misunderstood Wallace's post. However, $$\Omega_m$$ is already almost exactly analogous to $$h$$:

$$h = \frac{H_0}{100 km/s/Mpc}$$
$$\Omega_m = \frac{\rho_m}{\rho_c}$$

(Here $$\rho_c = \frac{3}{8 \pi G} H_0^2$$ is the critical density, the density for which k = 0 at a given expansion rate).

So we see that $$h$$ is the "true" Hubble constant compared against some "standard" value of 100 km/s/Mpc, while $$\Omega_m$$ is the "true" matter density compared against the "standard" density: the critical density. This is usually thought of as the density fraction, but that interpretation is only accurate if the curvature is zero.

 

[Petronut]

Ah, after reading this, I realize that I misunderstood Wallace's post. However, $$\Omega_m$$ is already almost exactly analogous to $$h$$:

$$h = \frac{H_0}{100 km/s/Mpc}$$
$$\Omega_m = \frac{\rho_m}{\rho_c}$$

(Here $$\rho_c = \frac{3}{8 \pi G} H_0^2$$ is the critical density, the density for which k = 0 at a given expansion rate).

So we see that $$h$$ is the "true" Hubble constant compared against some "standard" value of 100 km/s/Mpc, while $$\Omega_m$$ is the "true" matter density compared against the "standard" density: the critical density. This is usually thought of as the density fraction, but that interpretation is only accurate if the curvature is zero.

Very well stated, Chalnoth.