Approximate LCDM Expansion in Simplified Math (Part 4) - Comments

[Jorrie]

Jorrie submitted a new PF Insights post

Approximate LCDM Expansion in Simplified Math (Part 4)

Continue reading the Original PF Insights Post.

 

[marcus]

Hi Jorrie, glad you posted part 4!
Typo in the last paragraph before the endnotes.
Could delete the word "particularly" and say "one version"
or delete the "ly" and say "one particular version"

To tell the truth, modesty aside, it is a particularly *nice* version. But you can let people find that out for themselves : ^)

Congratulations on completing your Insight piece!

 

[marcus]

BTW I was wondering which galaxies we can see have always been receding > c
I suppose that most of the galaxies we can see are like that but I don't know the stretch cutoff.
Could be completely wrong about this---I have the idea that any galaxy we see at S > 2.9
and maybe actually at S>2.85 (something around there) has always been receding faster than c.

My reasoning is that if a galaxy is seen at S = 2.9 then using Lightcone7z its distance NOW is 0.97

And your sample galaxy ("generic") has a distance now of 14.4/17.3 = 0.832

So that means all the speed history of this other galaxy is scaled up by 0.97/0.832 = 1.166

and in particular the MINIMUM speed which always comes at time 0.44 is scaled up by 1.166
.8726 I think is the minimum speed for the sample galaxy so I have to multiply .8726*1.166 and it comes out
just a bit above 1.
I'm using too many decimal places and not taking the time to be neat, but this seems to show that any galaxy we see with S=2.9 or larger has always been receding faster than light. Not sure why one would even want to know that. : ^)

 

[PAllen]

I should also add that the closest SR analog of a cosmological recession rate is a celerity: a growth in proper distance from an 'observer', measured in some foliation of spacetime, by proper time of the moving object. This can be any factor greater than c in SR. If the astronomer does as is done with relative velocity in SR, and multiplies the proper time of the distant galaxy by the time dilation factor implied by the red shift (before dividing into the distance), you would get a less than c relative speed. More precisely, the GR analog of SR relative velocity is to parallel transport 4-velocities to a common event and compare them (take their scalar product per the metric; this gives gamma corresponding to relative velocity). While in GR, this result is path dependent it is ALWAYS < c. If the path you choose is that of the light, you find that the relative velocity is EXACTLY as implied by the redshift per SR doppler formula.

 

[PAllen]

Another observation is that a galaxy with recession velocity > c, has recession velocity always less than light emitted away from us by that galaxy. Thus it is always moving away from us slower than light is moving away from us. I think the both the value of recession velocity would be increased and artificial mystery removed, if it were more widely recognized that its SR analog is celerity not relative velocity.

Another way to see that the SR limit of recession velocity is celerity is to look at the Milne model, which is the zero mass limit of FLRW solutions (the spacetime is exactly Minkowski spacetime with 'funny' coordinates). This has recession velocity proportional to distance with no upper limit. But here, you can see, exactly, that these recession velocities are SR celerities. Thus, again, the SR analog of recession velocity is celerity, which already has no upper bound in SR.

 

[Jorrie]

BTW I was wondering which galaxies we can see have always been receding > c
I suppose that most of the galaxies we can see are like that but I don't know the stretch cutoff.
Could be completely wrong about this---I have the idea that any galaxy we see at S > 2.9
and maybe actually at S>2.85 (something around there) has always been receding faster than c.

Thanks Marcus.

Yes, I get the condition as S>2.85 for galaxies that have always been receding in terms of proper distance at more than c.

For the farthest observed galaxies, around S=10, at D_now=1.77 lzeit, if I scale V_gen up like in your post, they never dropped below some 1.86c in recession rate.
[/QUOTE]

 

[Jorrie]

Good observations, thanks PAllen.

I agree that the recession rate is more like celerity than speed. One of the reasons why I try to stick to recession rate and avoid the term 'speed'.

 

[marcus]

I've been noticing in the research literature that the Hubble parameter occasionally gets referred to as the "Hubble rate". I can't say how often or which articles because I didn't think to make a note of it. I came across an instance sometime in the past 5 or 6 days. I hope more professional authors pick up that usage---seems more straightforward and accurate than "Hubble constant" or "Hubble parameter".

Have to admit I'm not all that careful or consistent in my use of words---I tend to explore different ways of saying things. Often not a model of good English. I appreciate Jorrie's consistency and conservatism in language. Recently I've been trying out thinking of Hubble rate H(t) as a "speed-to-size" ratio.
Because it is what you multiply the size of a distance by to get its current growth speed.
That's what the Hubble law says: v = HD

And that translates right away (in terms of the scale factor) to a' = H a

So we are always seeing (what probably is the closest thing to a *definition* of H) that H = a'/a

which is known as the logarithmic derivative of a. The derivative of ln a.

Somehow our ordinary everyday language doesn't have a comfortable word for "logarithmic derivative" or "speed to size ratio".

It is like an interest rate, but *continuously compounded* (not based on any fixed finite time interval). Basically just thinking out loud here, talking to myself. The Hubble rate is somehow not really at home in the English language. But it is an important useful idea. Maybe without consciously trying to we are helping to make it more at home.

Anyway it is a real pleasure having your Insights article all here. Much food for thought. Some day I will get around to saying "e-fold" in this context

 

[Jorrie]

I'm often guilty of the same slip-up, but the term 'Hubble parameter' is usually reserved for the dimensionless 'h', which is Ho in conventional units expressed as a fraction of 100 km/s/Mpc. The presently accepted value is hence h=0.679.

'Speed to size ratio' is quite cool and perhaps a little more understandable than my favorite "fractional expansion rate".

 

[marcus]

The kicker about speed-to-size ratio is the calculus fact that if you have any positive increasing function f(t) its speed-to-size ratio is automatically the derivative of its logarithm.

the moment you see f'/f you recognize that it is the derivative of ln f

This is where we may have a roadblock in communicating with newcomers. They may not be familiar with the idea of f'(t) the time derivative of f(t)---or the slope of the f(t) curve. Or they might not be familiar with the natural logarithm and its derivative.

I'll imagine for a moment trying to explain ( and "log" is a more comfortable notation than "ln" so let's write it that way.)
The derivative of log(x) is 1/x
A beginner can see that by looking at the slope of the log(x) curve. It does what so much else in nature does---it's steep at first and levels out, becoming much more gradual.

If the beginner accepts that the derivative of log(x) is 1/x
then by the chain rule the derivative of log(f(t)) must be $\frac{1}{f(t)}f'(t)$ or f'/f
the slope of the nested thing is the product of the two slopes multiplied together.

So the Hubble rate, the speed-to-size ratio which nature gives us, is automatically the slope of the log of the scale factor a(t).

We can't get away from this. It's how nature is. So we have a hard explaining job. Either the beginner has to assimilate something that could be difficult and unfamiliar or we will be leaving an awkward gap in the picture. In any case that's my outlook at the moment. Can this be made palatable or must it be avoided? I don't suggest that it is a natural part of your essay. But it might be part of a teaching plan that covers basic cosmology, and which might use your Insights essay or something like it...

From another standpoint, I'd guess that in the view of many of the regulars at this forum, this is all trivial and it is absurd to even talk about it
: ^)

 

[marcus]

Here's an illustration of what I'm talking about. The cosmic scale function, call it a(t). This is the version that is not normalized to equal 1 at present (t = 0.8) so it equals slightly over 1.3 instead, but that does not matter for this. It's the blue curve.
And the LOG of the scale function--the red curve.
And the Hubble rate which is the SLOPE of the log of the scale function

You can see that at first the red curve (the log) rises very steeply, so the green curve (its slope) is high.
But then between 1 and 1.5 the red curve slope is almost down to a mere 1. You can read that off the graph. So the green curve which gives the slope settles down to near 1.
Between 1.5 and 2 the green curve is almost indistinguishable from constant 1, so the red curve has become virtually indistinguishable from a straight line with slope 1.

 

[marcus]

The green curve shows the real universe's actual Hubble rate (in zeit units). Just look where the green curve is at t=0.8.
You can check that at the present (t=0.8) it is 1.2 zeit^-1 ---which is right, corresponds to recent measurements of the Hubble rate. Incidentally the same as (0.83 zeit)^-1---0.83 zeit is the current value of the Hubble time and 0.83 lightzeit is the Hubble radius. Just have to take the reciprocal of 1.2.

 

[marcus]

In the hypersine thread I was wondering how one could address a newcomer's questions such as how do we know 0.8 (the expansion age)? and how do we know 2.67 lightzeit (the observable range now---presentday particle horizon)?
It seems like a lot, perhaps most, basic cosmology observational data falls along the blue curve: scalefactor-distance curve.

Incoming light tells you its source scalefactor (relative size of distances back when the light was emitted).
And then there are various opportunities to determine the presentday distance to its source.
So based on reasonable assumptions like GR describes gravity one draws the blue curve and fits it to the data and right away it points to 2.67.
That seems fairly straightforward, although it's complicated down at the level of detail.

But how to explain that the expansion age is 0.8 zeit?

It could be that I'm asking how to make it intuitive that we can add a third curve (time) to the graph, just reaching 0.8 at the right edge where a = 1.0.

https://www.physicsforums.com/insights/approximate-lcdm-expansion-simplified-math-part-4/

 

[marcus]

Or maybe the third curve should not be time (as function of scalefactor) but H(a) the expansion speed-to-size ratio (again as a function of a)

To recap, distance to source, and scalefactor, are directly measured, that's the blue curve---which multiplying by scalefactor immediately give the red curve. So we can consider those two as observed. Now the question is what can we derive. Can we derive H(a) the yellow curve? For sure, cosmologists do that by fitting the Friedmann model to the data, but can we make the chain of inference intuitive?

 

[Jorrie]

But how to explain that the expansion age is 0.8 zeit?

It could be that I'm asking how to make it intuitive that we can add a third curve (time) to the graph, just reaching 0.8 at the right edge where a = 1.0.

https://www.physicsforums.com/insights/approximate-lcdm-expansion-simplified-math-part-4/

I think Fig. 2.2 (https://www.physicsforums.com/insights/approximate-lcdm-expansion-simplified-math-part-2/) is the best to show the 0.8 zeit present time.

But more curves can be added, as long as it does not interrupt the flow of the Insight posts too much. Perhaps it is better to just use additional curves in answers to questions?