The Hubble constant − constant in what way?

[JimJCW]

TL;DR Summary

The general form of Hubble’s law is given by, v = H(t)D, where v is the recession velocity of an object, D is the object’s proper distance from our location, and H(t) is the Hubble parameter at cosmological time t. The value of H(t) at the present time, H(now), is called the Hubble constant. Some people consider this term as a misnomer because H(t) is t-dependent. Let’s plot the H vs. t curve with Jorrie’s calculator and discuss the result. We can share our understandings and questions.

The general form of Hubble’s law for a given cosmological time t is given by,

v = H(t)D, (1)​

where v is the recession velocity of an object, D is its proper distance, and H(t) is the Hubble parameter at t. To get the H vs. t plot based on the ΛCDM model, we can use the following steps:

• Start Jorrie’s calculator: http://jorrie.epizy.com/Lightcone7-2021-03-12/LightCone_Ho7.html?i=1. It uses PLANCK Data (2015) as default input.

• Keep the default values of z(upper) and z(lower).

• Click Open Column Definition and Selection, keep only Cosmic Time and H(z) selected, and click Open Column Definition and Selection again to close it.

• Select Chart and click Calculate. The H vs. t plot will appear.

• Note: You need to use Open Chart Options to adjust the ranges of the x- and y- coordinates to get the following plot:

As can be seen from the above figure, H(t) had very large values during the early times of the universe. We can use Jorrie’s calculator to get the following values (1 pc = 3.26 light year):

For the present time, the Hubble law becomes,

v = H(13.8) D = 67.74 D, (2)​

where H(13.8) = 67.74 km/s/Mpc is called the Hubble constant. This is probably because Eq. (2) is applicable for all D values at the present time. As an example, we can calculate v for an object near the edge of the observable universe, D = 46.5 Gly. Eq. (2) gives its recession velocity as v = 3.22 c.

@Jorrie
@Bandersnatch

 

[Will Learn]

@JimJCW
It is "constant" in that it doesn't depend on direction or location. All parts of space should obey that simple law regardless of which point in space you want to measure v and D from.

The change with respect to time is slow - by human standards. It would not have been obvious when the recession was first observed.

That's all I ever imagined when H was referred to as a constant.

 

[JimJCW]

@JimJCW
It is "constant" in that it doesn't depend on direction or location. All parts of space should obey that simple law regardless of which point in space you want to measure v and D from.

I think we all agree that the Hubble parameter H(t) is not a constant; it is t-dependent. Calling H(now) = 67.74 km/s/Mpc the Hubble constant, however, is considered as a misnomer by some. Your explanation in that case makes sense.

By the way, do you think the Hubble law is still valid if space does not expand? In that case, v = 0 for all D values. (just being curious)

 

[Ibix]

By the way, do you think the Hubble law is still valid if space does not expand? In that case, v = 0 for all D values. (just being curious)

As far as I'm aware the only way we know of to have a non-expanding universe is to carefully balance the matter and cosmological constant (Einstein's static universe). That results in no redshift (so was falsified by Hubble's observations), which you could argue satisfies Hubble's law with $H=0$.

 

[JimJCW]

As far as I'm aware the only way we know of to have a non-expanding universe is to carefully balance the matter and cosmological constant (Einstein's static universel. That results in no redshift (so was falsified by Hubble's observations), which you could argue satisfies Hubble's law with $H=0$.

The expansion of space presents a complex picture of the universe, for example, the H vs. t plot discussed here and Journey of an Observed Cosmic Microwave Background Photon posted earlier. I am hoping to find a model that does not involve expansion of space and can explain observations such as the redshift-distance relation and the origin of the CMB radiation. Do you know any?

Hubble's observations were about a redshift-distance relation for small z values. Hubble converted it to a velocity-distance relation by assuming that the redshift was caused by Doppler effect. It seems transforming Hubble's relation at that time to the present-day Hubble’s law applicable for the whole universe is a big extrapolation.

 

[PeroK]

Hubble's observations were about a redshift-distance relation for small z values. Hubble converted it to a velocity-distance relation by assuming that the redshift was caused by Doppler effect. It seems transforming Hubble's relation at that time to the present-day Hubble’s law applicable for the whole universe is a big extrapolation.

There have been nearly one hundred years of observations and data analysis since Hubble. And, several alternatives to the expanding universe have been investigated. It isn't the case that Hubble said something a hundred years ago based on minimal data and everyone since has believed it without question.

Not only that, but General Relativity predicted an expanding universe. Even Einstein, however, didn't have the courage to trust his own theory and fudged it away only later to realize his blunder.

Maybe the current cosmological models have a lot more going for them than a bit of extrapolation.

 

[weirdoguy]

I am hoping to find a model that does not involve expansion of space and can explain observations such as the redshift-distance relation and the origin of the CMB radiation. Do you know any?

Um, did you read what you quoted? It answers exactly what you asked:

As far as I'm aware the only way we know of to have a non-expanding universe is to carefully balance the matter and cosmological constant (Einstein's static universe). That results in no redshift (so was falsified by Hubble's observations),

It seems transforming Hubble's relation at that time to the present-day Hubble’s law applicable for the whole universe is a big extrapolation.

It would be a big extrapolation if nothing ever happened in between. You just threw out most of the cosmology...

 

[JimJCW]

There have been nearly one hundred years of observations and data analysis ...

And, several alternatives to the expanding universe have been investigated.

Please give us some examples of observations and investigations you are talking about. Are the observations specifically related to the Hubble law? What are the conclusions of the investigations? Does any of the alternatives have potentials? Do you like any of them?

Ibix specifically mentioned Hubble's observations. I tried to point out that those observations are for the redshift-distance relation, not velocity-distance relation directly. How about the observations you mentioned?

 

[JimJCW]

Um, did you read what you quoted? It answers exactly what you asked: ...

If we limit ourselves to Einstein’s field equation and the ΛCDM model, the question may not be interesting. I am interested in getting information on alternative models.

 

[PeroK]

Please give us some examples of observations and investigations you are talking about.

The observations that are the basis for modern cosmology. You could start here:

https://arxiv.org/abs/1204.5493

 

[JimJCW]

Let’s plot . . . with Jorrie’s calculator and discuss the result.

Let’s make more calculations about the Hubble parameter. To get the H vs. z plot based on the ΛCDM model, we can use the following steps:

• Start Jorrie’s calculator: http://jorrie.epizy.com/Lightcone7-2021-03-12/LightCone_Ho7.html?i=1. It uses PLANCK Data (2015) as default input.
• Set z(upper) = 100 and z(lower) = 0.
• Click Open Column Definition and Selection, keep only Redshift (z) and H(z) selected, and click Open Column Definition and Selection again to close it.
• Select Chart and click Calculate. The H vs. z plot will appear.
• Note: You need to use Open Chart Options to adjust the ranges of the x- and y- coordinates to get the following plot:

We can use Jorrie’s calculator to get the following values:

 

[Vanadium 50]

It is perhaps worth pointing out that Hubble's original observation was over a very small distance (2 MPc) compared to cosmological distances, as well as short times compared to cosmological times. On those scales, "constant" is an appropriate description, even if we know today that it is approximate.

It is not the only commonly used term that is more historical than precise. Cepheid variables are named after δ Cephei, even though that it is neither the first such star discovered, nor the nearest, nor the brightest. As fort Type-II Cepheids, there aren't any in Cepheus.

For that matter ω Centauri and BL Lacertae aren't even stars, even though they are named as such.

 

[JimJCW]

It is perhaps worth pointing out that Hubble's original observation was over a very small distance (2 MPc) . . .

You are right, as I mentioned in #5, Hubble's observations were about a redshift-distance relation for small z values, i.e., for small recession velocities and for small proper distances. The Hubble law for the present time is given by,

v = H(13.8) D = 67.74 D.​

The value H(13.8) = 67.74 km/s/Mpc is called a constant because now it is assumed to be applicable for all D values. As an example, we can calculate v for an object near the edge of the observable universe (at a proper distance of D = 46.5 Gly) to be 3.22 c. Maybe the Hubble constant is a constant in that way.

 

[Will Learn]

we can calculate v for an object near the edge of the observable universe

That is quite a resistant calculator you are selling, is it also water-proof?

 

[JimJCW]

That is quite a resistant calculator you are selling, is it also water-proof?

I like Jorrie’s calculator. It is almost water-proof; I found a discrepancy in the calculator earlier:

Discrepancy in Jorrie’s LightCone7-2017-02-08 Cosmo-Calculator​

I am studying cosmology on my own and need all the help I can get. I also use other calculators, such as those by Gnedin (now you need an account to access it) and Wright. Jorrie’s calculator is more convenient because of its tabular outputs and the chart function.

 

[JimJCW]

I also use other calculators, such as those by Gnedin (now you need an account to access it) and . . .

I communicated with Prof. Gnedin about it. He posted his calculator on another website so anybody can access it:

https://astro.uchicago.edu/~gnedin/cc/​

 

[Jorrie]

The value H(13.8) = 67.74 km/s/Mpc is called a constant because now it is assumed to be applicable for all D values. As an example, we can calculate v for an object near the edge of the observable universe (at a proper distance of D = 46.5 Gly) to be 3.22 c. Maybe the Hubble constant is a constant in that way.

Yes, this is how I would interpret it too. Constant for all comoving distances. And since comoving distance does not change with time (for us), Ho does not change over time for us.
It does however change whenever better data results in a new value for Ho.

 

[George Jones]

Some people consider this term as a misnomer because H(t) is t-dependent.

If quantity A is constant, this means that A doesn't change as some other quantity, say B, changes, and we say "A is constant with respect to B." Often, B is time, but B does not have to be time.

The Hubble constant is defined as $H\left(t\right) = \dot{a}\left(t\right)/a\left(t\right)$, where $a\left(t\right)$ is the scale factor of the universe, and $\dot{a}\left(t\right)$ is the rate at which the scale factor changes with respect to time. This means that, in general, $H\left(t\right)$ changes as the cosmic time $t$ changes.

$H\left(t\right)$, however, is constant in space, at any instant of cosmic time. If an instant of cosmic time $t$ is fixed, then three degrees of freedom are left in 4-dimensional spacetime. Call this space, and take this to the B of my first paragraph.

This is a reflection of the fact that Friedmann-Lemaitre-Robertson-Walker universes are homogeneous and isotropic in space, but not time.

 

[JimJCW]

The Hubble constant is defined as $H\left(t\right) = \dot{a}\left(t\right)/a\left(t\right)$, where $a\left(t\right)$ is the scale factor of the universe, and $\dot{a}\left(t\right)$ is the rate at which the scale factor changes with respect to time. This means that, in general, $H\left(t\right)$ changes as the cosmic time $t$ changes.

$H\left(t\right)$, however, is constant in space, at any instant of cosmic time. If an instant of cosmic time $t$ is fixed, then three degrees of freedom are left in 4-dimensional spacetime. Call this space, and take this to the B of my first paragraph.

According to Ryden’s Introduction to Cosmology,

. . ., the time-varying function H(t) is generally known as the “Hubble parameter,” while Ho, the value of H(t) at the present day, is known as the “Hubble constant.”​

You are saying that, in general, H(t) changes with cosmic time t, but at any given t, H(t) is a constant with respect to space, for example Ho. I agree.

You brought up an interesting expression,

The a vs. t and da/dt vs. t plots are shown below:

Since both a(t) and da/dt increase with time, one may want to know whether H(t) decreases or increases toward the future. As can be seen from the first figure of this post, it continually decreases with time and approaching an asymptotic value determined by ΩΛ, as a(t) becomes large,

H(∞) is calculated to be 56.3 km/s/Mpc using PLANCK Data (2015).