The awkward expression 1 km/s per Mpc-the radius is better

[marcus]

The awkward expression 1 km/s per Mpc--the radius is better

It's been pointed out that by historical accident we've been stuck with an awkward mix of units conventionally used to express Hubble parameter. If you parse the quantities km/s per Mpc you find that the length units cancel and you end up with the reciprocal of time.

So let's take one over the unit and see what time quantity it amounts to.

Go to google and type or paste this in:
1/(1 km/s per Mpc)

Of course you get a time, because the Hubble parameter unit is a reciprocal time in disguise, but it comes out in seconds!

However you can force the google calculator to give the quantity to you in years, simply by adding the words "in years" to what you paste into the window:

1/(1 km/s per Mpc) in years

Google will immediately give you: 977.8 billion years.

So that is what the unit boils down to: (km/s per Mpc) = (977.8 billion years)^-1
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But in most of the quantitative cosmology stuff we do around this forum we don't use parsecs and Megaparsecs very much, and recession speed are rarely expressed in km/s. So both those units are rather foreign to the discussion.

The problem, then, is to get the all-important Hubble parameter into some handier form for our purposes.

All things considered, I think the most user-friendly, in particular beginner-friendly, is the conventional Hubble radius. At any given stage in history, the Hubble radius is the distance that is expanding at speed c. So, because of the basic v = H D law, in whatever units you work in it is always true that c = H R and so the Hubble radius R = c/H.

Therefore, if somebody tells you that the Hubble parameter is, say, 67.8 km/s per Mpc,
all you have to do is paste this verbatim into google:

c/(67.8 km/s per Mpc)

and you will get out a certain distance. But it will be in meters! So to force google to tell you the Hubble radius in light years you simply add the words "in light years" to what you paste in:

c/(67.8 km/s per Mpc) in light years

Then google will give you 14.4 billion lightyears.

=======================

That is easy to interpret as a percentage distance growth rate namely 1/144 percent per million years. You can always do that with the Hubble radius if you want to explain it to someone as a percentage growth rate.

And it also means that a sample distance of 1 Gly is growing at a speed of 1/14.4 c, that is slightly less than 1/14 of the speed of light. All the other growth speeds are proportional, so that is also a good way to picture it.

In the standard LCDM model, at any given time, all the cosmological distances (i.e. between pairs of CMB stationary observers) are growing at the same percentage rate, and what seems to be the most user-friendly way to package that information is in the form of the Hubble radius.
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If you look at one of Jorrie's tables you can see what the Hubble radius has been in the past and what it is projected to be in the future. It has been increasing throughout history and is slated to continue increasing (but more and more gradually). So as it increases from the present 14.4 billion LY to, say, a future 17.3 billion LY, you can see that the percentage rate will decline from 1/144 to 1/173 percent per million years. You can always read off the percent rate from the Hubble radius.

 

[Jorrie]

If you look at one of Jorrie's tables you can see what the Hubble radius has been in the past and what it is projected to be in the future. It has been increasing throughout history and is slated to continue increasing (but more and more gradually). So as it increases from the present 14.4 billion LY to, say, a future 17.3 billion LY, you can see that the percentage rate will decline from 1/144 to 1/173 percent per million years. You can always read off the percent rate from the Hubble radius.

Marcus, I have also been using this calculation in the past, but since I gave done the sums for my TabCosmo9 extra expansion rate column, I'm no longer sure that this is valid. The % expansion rate per time unit (whatever that unit is) must be increasing exponentially in the far future, while the Hubble radius flattens out and tends to a maximum. Maybe I'm missing something, but I can't see it...

In any case $da/dt = a H(t)$ must be increasing without limit, not so?

 

[marcus]

...
In any case $da/dt = a H(t)$ must be increasing without limit, not so?

that's right. the equation is true by definition. And if you imagine that H(t) is constant then you have the usual recipe for exponential growth:

da/dt = a (the equation that e^t satisfies.)
or more generally da/dt = Ka, with some constant K

 

[marcus]

Constant H(t), or equivalently constant R(t) is the exact condition for ordinary exponential growth of a(t). Because H = a'/a so if it is constant there must be a constant proportion between the scale factor a(t) and its derivative a'(t). (I can't easily make the dot on a, so I use the prime )

======================

Talking about H(t) reminds me. a'(t) is not what I would call a percentage or fractional growth rate. Remember that a(now) is normalized to equal one so that sets the scale of a(t). Now if a'(t) = 0.007 per unit time when a(t) is SMALL that means much more growth percentage wise than a'(t) = 0.007 would mean at a time when a(t) is large. So a'(t) is not telling a percentage or fractional growth rate comparing it with the current a(t) value. It is more of an absolute growth rate.
======================

Also talking H(t) puts me in mind of something else.

As a physical quantity c/(67.9 km/s per Mpc) is the same as 14.4 Gly, just evaluated in different units. I swear by the google calculator in this matter

If you paste in "c/(14.4 billion lightyears) in (km/s per Mpc)" you get "67.90638.. km/s per Mpc"

If you paste in "c/(67.9 km/s per Mpc) in light years" you get 14.4007821 billion light years.

Somehow I think we need to emphasis the straightforward physical equality c/H = R. The fact that they are the same physical quantity written in different units. It is the definition of the Hubble radius R. Probably should be clearly established, maybe in a tooltip.

 

[Jorrie]

Constant H(t), or equivalently constant R(t) is the exact condition for ordinary exponential growth of a(t).

OK, so it's the old confusion of exponential absolute growth rate da/dt = Ka happening with a constant percentage growth rate. :)

I have erred in the info-popup of my da/dt column and will fix it. Thanks Marcus.

 

[Chalnoth]

It's been pointed out that by historical accident we've been stuck with an awkward mix of units conventionally used to express Hubble parameter. If you parse the quantities km/s per Mpc you find that the length units cancel and you end up with the reciprocal of time.

So let's take one over the unit and see what time quantity it amounts to.

Go to google and type or paste this in:
1/(1 km/s per Mpc)

Of course you get a time, because the Hubble parameter unit is a reciprocal time in disguise, but it comes out in seconds!

However you can force the google calculator to give the quantity to you in years, simply by adding the words "in years" to what you paste into the window:

1/(1 km/s per Mpc) in years

Google will immediately give you: 977.8 billion years.

So that is what the unit boils down to: (km/s per Mpc) = (977.8 billion years)^-1
================

But in most of the quantitative cosmology stuff we do around this forum we don't use parsecs and Megaparsecs very much, and recession speed are rarely expressed in km/s. So both those units are rather foreign to the discussion.

The problem, then, is to get the all-important Hubble parameter into some handier form for our purposes.

All things considered, I think the most user-friendly, in particular beginner-friendly, is the conventional Hubble radius. At any given stage in history, the Hubble radius is the distance that is expanding at speed c. So, because of the basic v = H D law, in whatever units you work in it is always true that c = H R and so the Hubble radius R = c/H.

Therefore, if somebody tells you that the Hubble parameter is, say, 67.8 km/s per Mpc,
all you have to do is paste this verbatim into google:

c/(67.8 km/s per Mpc)

and you will get out a certain distance. But it will be in meters! So to force google to tell you the Hubble radius in light years you simply add the words "in light years" to what you paste in:

c/(67.8 km/s per Mpc) in light years

Then google will give you 14.4 billion lightyears.

This post doesn't make much sense to me. I could just as easily type in:

1/(67.8 km/s per Mpc) and it will give me 14.4 billion years. The conversion from time to distance literally just adds the word "light" in front of the units. And it's just as easy to interpret as a percent expansion per million years.

 

[Jorrie]

Also talking H(t) puts me in mind of something else.

As a physical quantity c/(67.9 km/s per Mpc) is the same as 14.4 Gly, just evaluated in different units. I swear by the google calculator in this matter

If you paste in "c/(14.4 billion lightyears) in (km/s per Mpc)" you get "67.90638.. km/s per Mpc"

If you paste in "c/(67.9 km/s per Mpc) in light years" you get 14.4007821 billion light years.

I could just as easily type in:

1/(67.8 km/s per Mpc) and it will give me 14.4 billion years.

The Google calculator's very smart interpretation of mixed units can sometimes be a hindrance to a beginner's understanding of what is going on. Without saying so, it brings in conversion factors as required: c = 3 x 10⁵ km/s, 1 Mpc = 3.26 x 10⁶ light years, and 1 year = 3.16 x 10⁶ seconds.

Taken at face value, a statement like "c/(67.9 km/s per Mpc) = 14.4 billion light years" is very confusing, so I think it is good that it gets explained. I prefer that for economy, just the conversion factor: 1 km/s per Mpc = 977.8 Gy^-1 be stated explicitly when we talk about R_H = c/H₀ and use conventional units.

We all know about the spaceflight mishaps that have happened due to units mismatches...

 

[Chalnoth]

The Google calculator's very smart interpretation of mixed units can sometimes be a hindrance to a beginner's understanding of what is going on. Without saying so, it brings in conversion factors as required: c = 3 x 10⁵ km/s, 1 Mpc = 3.26 x 10⁶ light years, and 1 year = 3.16 x 10⁶ seconds.

Taken at face value, a statement like "c/(67.9 km/s per Mpc) = 14.4 billion light years" is very confusing, so I think it is good that it gets explained. I prefer that for economy, just the conversion factor: 1 km/s per Mpc = 977.8 Gy^-1 be stated explicitly when we talk about R_H = c/H₀ and use conventional units.

We all know about the spaceflight mishaps that have happened due to units mismatches...

I think it's nice to explain, but I was just pointing out that converting it to distance doesn't really help.

After all, the statement that the universe expands by 1/144th of a percent every million years comes directly from the statement that 1/67.9km/s per Mpc = 14.4 billion years. Basically, Marcus' explanation can be dramatically simplified by leaving R(t) out of it.

 

[marcus]

...
Taken at face value, a statement like "c/(67.9 km/s per Mpc) = 14.4 billion light years" is very confusing, so I think it is good that it gets explained. I prefer that for economy, just the conversion factor: 1 km/s per Mpc = 977.8 Gy^-1 be stated explicitly when we talk about R_H = c/H₀ and use conventional units.

We all know about the spaceflight mishaps that have happened due to units mismatches...

There was a very disappointing loss of a Mars mission due to human ineptitude at handling units. Its good to be reminded. Humility and care is called for on all sides. Units will often sneak up and bite one in the leg and this is a general feature of the human condition, I think.

In the above, alas, you probably mean (977.8 Gy)^-1 which is different from 977.8 Gy^-1 and it isn't really anyone's fault, the damn things are treacherous.

We all come to terms in different ways. We learn different habitual routines for getting along with units.

I had a very solid hard-head Midwesterner physics prof who believed in using expressions which equaled one. The way you converted complicated units expressions, he believed, was you inserted chunks of notation which you knew equaled one into the long string of bewildering gibberish and then you began canceling and it all became simple. He was the kind of practical physicist who got on famously with engineers.

Here's how he would have approached this, we have this funny unit "km/s per Mpc"
and we know that the following expression is identically equal to 1.

(977.8 billion years)km/s per Mpc = 1

You know that is identically one because you can patiently work it out. You would change the 977.8 billion years into seconds, and then the km/s turns that into a number of kilometers, and that turns out to be the number of kilometers in a Megaparsec. So then you have "Mpc per Mpc", which is identically unity. So what I said is unarguably true (he would stare very hard at you then). And so I can insert it into the algebra without changing anything.

Forever afterwards you know you can insert that into the string of quantities you are staring at without changing anything. Because putting a factor of one in never changes anything. I guess I'm being boring. I remember this guy with affection and respect. This was how he did it. Everybody learns their own tricks.

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My own (different) personal adaptation is that I sincerely believe in the reality of physical quantities as somethings completely apart from the NUMBERS we associate with them. The quantities are real, real speeds real lengths etc. And the numbers and units are just operational conventions.

So I would not say Taken at face value, a statement like "c/(67.9 km/s per Mpc) = 14.4 billion light years" is very confusing.
Indeed you could be right! It could be confusing to most people and this is a blind spot on my part. But to me that thing in quotes is exactly the point I would want to get across. They are indeed the identical same physical quantity. Hubble law is not about numbers but about quantities. Hubble law says v=HD and in this case v=c and D=R and so c=HR, and so it has to be true that c/H = R. It is the bedrock fact. It can't be confusing! How can it be confusing to say c/H = R?

But maybe we do need to introduce this factor (977.8 Gy) or its reciprocal (977.8 Gy)^-1 into the discussion. I trust your judgment and good sense, based on experience, so I'm going to try to be more moderate and less opinionated about this.

 

[marcus]

This post doesn't make much sense to me. I could just as easily type in:

1/(67.8 km/s per Mpc) and it will give me 14.4 billion years. The conversion from time to distance literally just adds the word "light" in front of the units. And it's just as easy to interpret as a percent expansion per million years.

I've been experimenting with rewording tooltips and I've taken a page from what Chally is saying here. I think he's saying it's ROUNDABOUT to work with distance when you can so easily explain things in terms of time.

I think freshmen and other greenhorns should be introduced to google calculator as soon as possible because it is a real innovation in convenience to have a calculator that knows most of the usual physics constants. So I'm looking for opportunities to put a straightforward google-ator application into a tooltip.

Chally says: "...1/(67.8 km/s per Mpc) and it will give me 14.4 billion years..."

That is simple and straightforward, so I want to look at the tooltip for Hubble constant that says:

==suggested text==
At any stage in expansion history the Hubble rate H and the Hubble radius R are related by c = HR. Conventional notation for the "now" Hubble rate is H_o. It is uniquely determined by the requirement that it give the correct "now" Hubble radius, in other words we must have R_now = c/H_o. As a check, paste 1/(67.9 km/s per Mpc) into google and press = to activate the calculator. You should get 14.4 billion years, so multiplying by c would give the correct 14.4 billion lightyears for R_now.
==end==
Hope this is not too long-winded. See what you think.

 

[Chalnoth]

Well, the ultimate purpose of this post is to try to understand the meaning of the Hubble expansion rate, correct? I could think of two ways of doing it. The first is to look, like you do, at the expansion rate as a percent expansion per unit time. I think that's perfectly-good, and can be described in relatively few words.

A second description that I think works well is to think of the expansion rate as compared to the typical velocities that galaxies have relative to one another. Typically, galaxies will have peculiar velocities that are less than around 1000km/s relative to other nearby galaxies. The highest velocities happen in the most massive galaxy clusters. So a natural question to ask is: how far away do two galaxies have to be for their relative velocity due to the expansion be 1000km/s. Since v = Hd, that's about (1000km/s) / (67.8km/s/Mpc) = 14.7Mpc. You can have Google convert this quantity to light years if you like (it's about 48 million light years). But this shows that once you have separations that are greater than around 48 million light years, the expansion velocity is dominant compared to the so-called peculiar velocity (the velocity of the galaxy relative to the background: ours is about 600km/s, by the way).

It might then be worthwhile to start talking about what 14.7Mpc actually means in terms of distance. You could compare this to the sizes of typical galaxies, or to the typical distances between galaxies, or to the sizes of clusters of galaxies or larger structures.

Either way, I do think it's valuable to talk about multiple ways of describing the exact same physical phenomenon. Some people understand one way of understanding something better than another, and it's also worth pointing out that there are many valid perspectives.

 

[Mordred]

.

Either way, I do think it's valuable to talk about multiple ways of describing the exact same physical phenomenon. Some people understand one way of understanding something better than another, and it's also worth pointing out that there are many valid perspectives.

Thats a good point,

 

[marcus]

I want to revisit this equality

...
(977.8 billion years)km/s per Mpc = 1

You know that is identically one because you can patiently work it out. You would change the 977.8 billion years into seconds, and then the km/s turns that into a number of kilometers, and that turns out to be the number of kilometers in a Megaparsec. So then you have "Mpc per Mpc", which is identically unity...

Another way to check that, of course, is to go to google and paste in
(977.8 billion years)km/s per Mpc
and press = to activate the google calculator. It will say something like 0.9999..., essentially one.

We know that the Hubble time is 1/H by definition. So what this is telling us is that the Hubble time corresponding to a Hubble rate of 1 km/s per Mpc is 977.8 billion years.

That may seem useless to you at first sight but it is really very helpful (Jorrie first demonstrated that in this PF context, as far as I know.)

What it means is that if someone gives you a Hubble time quantity like 17.3 billion years and for some reason you want to see what RATE (in km/s per Mpc) it corresponds to, all you need to do is divide 977.8 by 17.3 .

Because the smaller the Hubble time the faster the rate. So you calculate 977.8/17.3 = 56.5
and that's it: 56.5 km/s per Mpc is the rate that corresponds to the time 17.3 billion years.

You know it has to be that once you realize that 977.8 billion years is the time corresponding to a rate of ONE km/s per Mpc.

977.8/2 billion years is the time corresponding to a rate of TWO km/s per Mpc.
977.8/3 billion years is the time corresponding to a rate of THREE km/s per Mpc.
977.8/56.5 billion years is the time corresponding to a rate of 56.5 km/s per Mpc
but that happens to be 17.3 billion years,...and so on.

And that extends to Hubble radius as well. 977.8 billion light years is the radius corresponding to the rate 1 km/s per Mpc. So if someone tells you a Hubble radius and for some reason you want to know the rate (expressed in km/s per Mpc) you can readily find it in exactly the same way.
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To take an example, some one tells you Hubble radius 14.4 billion light years.
You can immediately, without arithmetic, visualize the corresponding rate as 1/144 percent per million years.
But if you want you can also do the arithmetic 977.8/14.4 = 67.9, and say it in conventional units as 67.9 km/s per Mpc.