Effort to get us all on the same page (balloon analogy)

[marcus]

Ellis and Rothman, in their Am.J.Phys. paper "Lost Horizons", use the term "onion", and I think that I have seen this term used in a few other places.

From Lost Horizons:

I'm a big fan of George Ellis but I think he made a mistake in the produce department. Lightpear has fewer syllables and is more accurately descriptive than "lightonion".

Onions tend to be altogether too round, and how could one resist this depiction of an Anjou pear:
http://carrotsareorange.com/wp-content/uploads/2010/05/pear-anjou.jpg

However in any case one should never say lightcone, so I would approve switching to either vegetable of terminology.

 

[marcus]

Hi Marcus.

Rather than trying to find a "better explanation" for the angular diameter max, I have spent the time more fruitfully (I hope) to update my cosmo-calculator to include values on your latest table (plus some presentational enhancements). Have not substituted it on my website yet, but here is a temporary link for testing purposes. I have opted for a more conventional value for your H(d^{-1}), namely 'Time for 1% proper distance increase' in Gy, since it fits in better with my calculator's units and style. I hope I have the conversion correct?

I would appreciate comments from yourself and any other interested parties. In time I should also add some more descriptive notes/links.

Jorrie, as long as we are in a lighthearted mood and you will take any comments on calc as friendly intended, I'll make a few comments just from personal PoV. I think it is a great calculator and extremely useful.

However I would change "distance traveled by light" to "lookback time" and write My instead of Mly. I just think it is the more conventional term. What you are really talking about is the "light transit time"---the time it took for the light to get here. And people customarily call that the lookback time. I guess you could also call it "light transit time". or "light travel time". Not sure which is best.

I know what you mean by "distance traveled by light"---it's the distance it would have traveled on its own (without the help of expansion) but it's a bit confusing to refer to lookback time in those terms and also not conventional.

And also it would be more conventional (and slightly more correct mathematically) to say
"1% of Hubble time" and give units (as you do) in My (instead of time needed for 1% increase...).

Hubble time (defined as 1/H) is standard terminology and I think it's really nice to have the calculator give 1% of it in millions of years, because it is a good reminder of how I am always thinking of the instantaneous distance growth rate. So convenient! You just take the number that shows up in the box, e.g. 139.xxx, and write one over it, and bingo you have 1/139 % per My. A great way to visualize H as a distance growth rate!

That one excellent convenience outweighs my quibbles of terminology so I would be glad to see you make the changes "as is" based on that alone!

However since you asked for comment, i am quibbling that it would be conventional and slightly more mathly correct to say "1% of Hubble time" or maybe use the asterisk and label it "1% of Hubble time*"

where down below your footnote says something like "*approximate time needed for a 1% growth of proper distance"

You realize that a bank account that grows at the instantaneous rate of 1% per year (continuously compounded) will therefore grow slightly more than 1% in the course of a year. Strictly speaking you have to say "approx." because the reciprocal of an instantaneous rate, which is what the Hubble constant is, slightly understates the amount of growth in the given unit of time due to continuous compounding.

I hope you do write a few notes to accompany the calculator.

EDIT: I just put e^.01 into the google calculator and got 1.01005017
which is so close to 1% that I feel foolish making the distinction. If something grows at an instantaneous rate of 1% per million years, then if you wait 1 million years then (even with continuous compounding which is a feature of instantaneous rates) it will to any reasonable person look like it has grown 1%.
Why should I fuss about the difference between 1% and 1.005%. OK OK. No objection to the new version of your calculator. Go with it.

 

[George Jones]

I'm a big fan of George Ellis but I think he made a mistake in the produce department.

I hope that you will forgive me for straying a little further off topic. George Ellis, Roy Maartens, and Malcolm MacCallum have coauthored a new advanced cosmology text,

https://www.amazon.com/dp/0521381150/?tag=pfamazon01-20

I like Ruth Durrer's review. I have ordered a copy, which I should receive on Monday.

 

[marcus]

I hope that you will forgive me for straying a little further off topic. George Ellis, Roy Maartens, and Malcolm MacCallum have coauthored a new advanced cosmology text,

https://www.amazon.com/dp/0521381150/?tag=pfamazon01-20

I like Ruth Durrer's review. I have ordered a copy, which I should receive on Monday.

Looks to be an important book! Some additional information on this page:
https://www.amazon.com/dp/0521381150/?tag=pfamazon01-20
Nice cover illustration! Two little blobs of overdensity in the microwave skymap giving birth to a cluster of galaxies! Picture worth many words.

April 2012 Cambridge U.P. and browsing allowed at the Amazon page. I will have a look at the ToC. thanks for the pointer!

 

[Jorrie]

However I would change "distance traveled by light" to "lookback time" and write My instead of Mly. I just think it is the more conventional term.

I fully agree with lookback time as more conventional, but I thought the distance interpretation to be more intuitive than lookback time, which for beginners has to be explained. I guess that with more notes/footnotes, this requirement may however be met with the conventional term.

... it would be more conventional and mathly correct to say "1% of Hubble time" or maybe use a footnote and label it "1% of Hubble time*"
where down below your footnote says something like "*approximate time needed for a 1% growth of distance"

Excellent idea! Gives us "two for the price of one" in terms of info.

 

[marcus]

I hope that you will forgive me for straying a little further off topic. George Ellis, Roy Maartens, and Malcolm MacCallum have coauthored a new advanced cosmology text,

https://www.amazon.com/dp/0521381150/?tag=pfamazon01-20

I like Ruth Durrer's review. I have ordered a copy, which I should receive on Monday.

I took a peek at pages 526-530, the section called "20.4 Loop quantum gravity and cosmology"
Page 537: " Like string theory, loop quantum gravity is still in its infancy--and either or both of these candidate quantum gravity theories could fail as a result of further discoveries...
...Given the uncertain status of all current attempts to develop quantum gravity, it is also useful to have competing paradigms."

Starting on page 537 you get section 20.4.1 "Basic features of quantum geometry" which is a thumbnail sketch of LQG with its main results (discrete area spectrum, Immirzi parameter..)
On page 528 begins section 20.4.2 "Loop quantum cosmology"
followed by section 20.4.3 "Loop quantum cosmology resolution of the big bang singularity
ending on page 530 with Figure 20.4 showing the evolution of the scalefactor during the LQC bounce,
and giving the semiclassical modified Friedmann and Raychaudhuri equations (equations 20.44 and 20.45)

It's highly condensed but all in all pretty good!
Eqn 20.41 gives the density range where LQC differs from classical, namely
ρ ≥ ρ_Planck.
Eqn 20.42 gives an equation for the the critical density ρ_crit, the max density achieved at bounce, and says that under usual assumptions works it out to about 0.4ρ_Planck.
Eqn 20.43 indicates that an inflationary epoch would begin after a large expansion resulting from the bounce itself which reduces the density initially by a factor of 10^-11.
(ρ/ρ_crit)_infl~10^-11.

These are consequences of 20.44 and 20.45 which are the familiar Friedmann and Raychaudhuri equations with an addiitional term ρ/ρ_crit which is suppressed except at densities near Planck scale. The authors cover the basic LQC stuff that researchers working on LQC phenomology use regularly. Roy Maartens has written some Loop cosmology pheno papers as I recall. The treatment is brief but impresses me as thoroughly solid/knowledgeable. Glad to see it in a major advanced cosmology text like this!

To take this peek (in case anyone wants to) you just go to the Amazon page and click on "look inside" and enter "loop quantum gravity" in the search box. It will give you a choice of clicking on page 513 or 526. I happened to choose page 526. The other passage seems more general overviewy, so less interesting.

 

[marcus]

Jorrie, I didn't see your post #355 and edited my post #352 to remove a minor objection. So far everything you are proposing looks fine from here!

 

[marcus]

Since we've turned a page, I'll bring forward the earlier table, to have it handy. It shows the then-distance maximum around 5.8 billion ly. To remind anyone who happens to be reading, the numbers in this table were gotten with the help of Jorrie's calculator. The calculator gives multidigit precision and I've rounded off. Hubble rates at various times in past are shown both in conventional units (km/s per Mpc) and as fractional growth rates per d=10⁸y. The first few columns show lookback time in billions of years, and how the Hubble rate has been declining, while the Hubble radius (reciprocally) has extended out farther. The columns on the right show the proper distance (in Gly) of an object seen at given redshift z both now and back when it emitted the light we are currently receiving. The numbers in parenthesis are fractions or multiples of the speed of light showing how rapidly the particular distance was growing.

 Standard model with WMAP parameters 70.4 km/s per Mpc and 0.728. 
Lookback times shown in Gy, distances (Hubble, now, then) are shown in Gly.
The "now" and "then" distances are shown with their growth speeds (in c)
time      z     H(conv)   H(d[SUP]-1[/SUP])    Hub      now          back then 
   0     0.000     70.4   1/139    13.9      0.0          0.0
   1     0.076     72.7   1/134    13.4      1.0(0.075)   1.0(0.072)
   2     0.161     75.6   1/129    12.9      2.2(0.16)    1.9(0.14)
   3     0.256     79.2   1/123    12.3      3.4(0.24)    2.7(0.22)
   4     0.365     83.9   1/117    11.7      4.7(0.34)    3.4(0.29)          
   5     0.492     89.9   1/109    10.9      6.1(0.44     4.1(0.38
   6     0.642     97.9   1/100    10.0      7.7(0.55)    4.7(0.47)
   7     0.824    108.6   1/90      9.0      9.4(0.68)    5.2(0.57)
   8     1.054    123.7   1/79      7.9     11.3(0.82)    5.5(0.70)
   9     1.355    145.7   1/67      6.7     13.5(0.97)    5.7(0.86)
  10     1.778    180.4   1/54      5.4     16.1(1.16)    5.8(1.07)
  11     2.436    241.5   1/40      4.0     19.2(1.38)    5.6(1.38)
  12     3.659    374.3   1/26      2.6     23.1(1.67)    5.0(1.90)
  13     7.190    863.7   1/11      1.1     29.2(2.10)    3.6(3.15)
 13.6   22.22    4122.8   1/2.37    0.237   36.7(2.64)    1.6(6.66)

Abbreviations used in the table:
"time" : Lookback time, how long ago, or how long the light has been traveling.
z : fractional amount distances and wavelengths have increased while light was in transit. Arriving wavelength is 1+z times original.
H : Hubble expansion rate, at present or at times in past. Distances between observers at rest grow at this fractional rate--a certain fraction or percent of their length per unit time.
H(conv) : conventional notation in km/s per Megaparsec.
H(d^-1) : fractional increase per convenient unit of time d = 10⁸ years.
"Hub" : Hubble radius = c/H, distances smaller than this grow slower than the speed of light.
"now" : distance to object at present moment of universe time (time as measured by observers at CMB rest). Proper distance i.e. as if one could freeze geometric expansion at the given moment.
"then" : distance to object at the time when it emitted the light.

Remember that "proper" distance, the distance used in Hubble law to describe expansion, is "freezeframe". The proper distance at a given moment in Universe time is what you would measure (by radar or string or whatever usual method) if at that moment you could stop the expansion process long enough to make the measurement.
The Hubble law describes the expansion of distances between observers at rest with respect to the background of ancient light and the process itself. Observers who see the ancient light and the expansion process approximately the same in all directions, e.g. no Doppler hotspots.
The field of an observer's view is not conical, but rather it is pear-shape because distances were shorter back then. Here is a picture of an Anjou pear.
http://carrotsareorange.com/wp-content/uploads/2010/05/pear-anjou.jpg
Here is Lineweaver's spacetime diagram:
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure1.jpg
The upperstory figure, with horizontal scale in proper distance, shows the lightpear outline.
Here is Lineweavers plot of the growth of the scalefactor R(t), which models the growth of all distances between observers at universe-rest (at rest with respect to background.)
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure14.jpg
The dark solid line is according to standard model parameters. Various other cases are shown as well.

 

[marcus]

... it would be more conventional and mathly correct to say "1% of Hubble time" or maybe use a footnote and label it "1% of Hubble time*"
where down below your footnote says something like "*approximate time needed for a 1% growth of distance"

Excellent idea! Gives us "two for the price of one" in terms of info.

The "Hubble time" 1/H is an interesting variable. Normally we write the core cosmology equations in terms of H, but "time needed for a 1% growth of distance" is intuitively appealing. Let's try, as an experiment, writing the Friedmann eqn (flat case, matter era) in terms of Y = 1/H, instead of in terms of H.

I will show that the Friedmann equation translates into a very simple differential equation for Y:

Y' = (3/2)[1- (Y/Y_∞)²]

A nice feature is that Y is a time so its derivative (ΔY/Δt) is a pure number.
We can easily see that the present value of this number is 0.4

In other words it doesn't matter what unit of time we use. The units cancel. So for example let me use the cosmologically convenient time unit d = 10⁸ year.

Y = 139 d
Y_∞ = 163 d
Y' will be the increase in Y per unit time.
Y' = (3/2)[1- (139/163)²] = 0.41... ≈ 0.4

So over the next 100 million years we can expect an increase from 139.0 to 139.4.

In other words, if a one percent increase in distance NOW takes 139 million years, looking ahead it will then take 139.4 million years.

Or if you think of the Hubble growth rate as now being 1/139 percent per million years, at that time in the future it will have decreased to 1/139.4 percent per million years.

Of course in the very long run we know that H will settle down to 1/163 percent per million years, but it's nice having a simple differential equation for Y so one can how it is changing at present, on what timescale.

 

[marcus]

For people who would like to see the (elementary calculus) way the equation for Y' is derived:

Y' = (1/H)' = - H'/H² = 4πGρ/H² = 4πGρY²

All this uses is H' = - 4πGρ, which we know from a previous post. And then we use the Friedmann equation to get an expression for ρ, and substitute it into the above.

ρ = (3/8πG)[1/Y² - 1/Y_∞²]

Y' = (3/2)[1/Y² - 1/Y_∞²]Y²
= (3/2)[1- Y²/Y_∞²]

The square of the ratio 139/163 is a familiar model parameter that is often quoted, namely 0.728.
Here we give it a new significance as determining the current rate of increase of the Hubble time.
One minus 0.728, namely 0.272, multiplied by 3/2, is this number 0.41... we're talking about.

The current value of the Hubble time is increasing 0.4 year per year. Or 0.4 century per century. Or 0.4 Gy per Gy. That is (if the rate were steady it would result in) an increase from 13.9 billion years to 14.3 billion years in a billion year interval. find it more convenient to think of time in units of d. So I say the Hubble time is increasing 0.4 d per d---or from 139.0 to 139.4 d in 100 million years.

Since the equations here are based on introductory work in post#313, which is several pages back, I will bring forward part of that post:

=====quote post#313======
By definition H = a'/a, the fractional rate of increase of the scalefactor.

We'll use ρ to stand for the combined mass density of dark matter, ordinary matter and radiation. In the early universe radiation played a dominant role but for most of expansion history the density has been matter-dominated with radiation making only a very small contribution to the total. Because of this, ρ goes as the reciprocal of volume. It's equal to some constant M divided by the cube of the scalefactor: M/a³.
Differentiating, we get an important formula for the change in density, namely ρ'.
ρ' = (M/a³)' = -3(M/a⁴)a' = -3ρ(a'/a) = -3ρH
The last step is by definition of H, which equals a'/a

Next comes the Friedmann equation conditioned on spatial flatness.
H² - H_∞² = (8πG/3)ρ
Differentiating, the constant term drops out.
2HH' = (8πG/3)ρ'
Then we use our formula for the density change:
2HH' = (8πG/3)(-3ρH) = - 8πGρH, and we can cancel 2H to get the change in H, namely H':

H' = - 4πGρ
====endquote====

That's all we needed from the earlier post, but I'll quote the rest of the passage to have it handy. What follows is extra: This is a derivation of the so-called "Raychaudhuri" or "Second Friedmann" equation. Also called the "acceleration Friedmann equation" because it gives a handle on the second derivative of the scalefactor a(t).

===continuation===
Again by definition H = a'/a so we can differentiate that by the quotient rule and find the change in H by another route:
H' = (a'/a)' = a"/a - (a'/a)² = a"/a - H²

Now the Friedman equation tells us we can replace H² by H_∞² + (8πG/3)ρ. So we have
H' = a"/a - H² = a"/a - H_∞² - (8πG/3)ρ = - 4πGρ

We group geometry on the left and matter on the right, as usual, and get:
a"/a - H_∞² = (8πG/3)ρ - 4πGρ = - (4πG/3)ρ
using the arithmetic that 8/3 - 4 = - 4/3

This is the socalled "second Friedmann equation" in the matter-dominated case where radiation pressure is neglected.
a"/a - H_∞² = - (4πG/3)ρ
In the early universe where light contributes largely to the overall density a radiation pressure term would be included and, instead of just ρ in the second Friedmann equation, we would have ρ+3p.
===endquote===

 

[marcus]

Let's continue the experiment of using the Hubble time Y = 1/H as a variable. We have a simple differential equation for Y:
Y' = (3/2)[1- Y²/Y_∞²]
so if we specify initial conditions we can plot Y(t) over some time interval, say from t1 to t2.

Now suppose we want to calculate the redshift corresponding to that interval. All we have to do is numerically integrate 1/Y: add up values of 1/Y along that interval.

∫_t1^t21/Y dt = log(1+z)
===================

We can take a simple example to see how this works. Using the differential equation for Y, we have that Y'(now) = (3/2)[1- 139²/163²] = 0.4
Let's take as interval the last 100 million years, with t2 being the present.
So we can estimate Y(t1) = 138.6 d and Y(t2) = 139.0
The average value of 1/Y in that interval we can estimate to be
(1/138.6 + 1/139)/2 = 0.0072046
The interval is one timeunit long so we can take that to be the value of the integral, and that is therefore the log of 1+z.
So we take exp(0.0072046) and we get 1+z ≈ 1.00723.
So z ≈ 0.00723
As a check, I should be able to put that in Jorrie's calculator and get that the lookback time is 100 million years.
Pretty close: 99.96 million years.

As another example I went back 300 million years, with the differential equation for Y and got these values:
139, 138.59, 138.17, 137.78
Then performed a rough numerical integration using google calculator and took exp:
exp(1/139/2+1/138.59+1/138.17+1/137.75/2)
The result was 1+z ≈ 1.0219,
so z ≈ 0.0219
As a test, putting that into J's calculator we get that the lookback time is 299.59 million years. Should be 300 but considering the quick and dirty numerical integration, not too bad either.

 

[marcus]

Using Y rather than H as the main variable seems to be working out. Here's another application:
First of all what was our Friedmann equation
H² - H_∞² = (8πG/3)ρ
now becomes
1/Y² - 1/Y_∞² = (8πG/3)ρ
and we can solve that for the density:
ρ = (3/8πG)[1/Y² - 1/Y_∞²]

Suppose we want to calculate the density (this is partly review) using google calculator. I'd like to specify that the units be in "nanopascal/c²"
A nanopascal works an energy density unit for me. A joule of energy per cubic kilometer. I know what a joule is: I can do a joule of work or make a joule of noise. And I know kilometers. So I can picture a joule of energy spread out in a cubic km. Dividing by c² just converts that from energy density to equivalent mass density.

So I'm going to make google calculator compute the current matter-and-radiation density of the universe and at the end I'll specify the answer should be "in nanopascal/c²". The calculator will know how to put the answer in those terms. See if this makes sense to you, and try pasting it into the google search box.

3/(8pi*G)(1/139^2 - 1/163^2)(percent per million years)^2 in nanopascal/c^2
Or equivalently, paste in:
3/(8pi*G)(1/139^2 - 1/163^2)/(10^8 years)^2 in nanopascal/c^2
or:
3/(8pi*G)((139e8 years)^-2 - (163e8 years)^-2) in nanopascal/c^2

All give the same result: 0.2279 nanopascal/c²

As far as we know, something like that, something around 0.23 nanopascal/c²,
is the combined mass density of all the ordinary matter, dark matter, and radiation, in the universe.
(I'm treating the cosmological constant simply as a residual curvature constant intrinsic to spacetime, not associated with any fictitious energy. So the mass density represents the total of stuff we actually know about--dark matter is real enough, astronomers are mapping its concentration in different parts of space, although the actual particles have not been definitively detected so far.)

 

[Jorrie]

All give the same result: 0.2279 nanopascal/c²

As far as we know, something like that, something around 0.23 nanopascal/c²,
is the combined mass density of all the ordinary matter, dark matter, and radiation, in the universe.
(I'm treating the cosmological constant simply as a residual curvature constant intrinsic to spacetime, not associated with any fictitious energy. So the mass density represents the total of stuff we actually know about--dark matter is real enough, astronomers are mapping its concentration in different parts of space, although the actual particles have not been definitively detected so far.)

Hi Marcus; a very interesting view and discussion. I'm used to treat the cosmological constant as some energy of the vacuum, the energy density of which remains constant with expansion (more volume, more energy, linearly). I think your view is equivalent and in a way more accessible than the standard view - it seems to remove the difficulties of explaining the negative pressure and the way it balances the energy of the 'Friedman books' for flatness...

I have completed an updated cosmo calculator for test purposes, with some 'info buttons' to make it more palatable for inexperienced users. When you have time, please take a peek at the info button contents.

 

[marcus]

...
I have completed an updated cosmo calculator for test purposes, with some 'info buttons' to make it more palatable for inexperienced users. When you have time, please take a peek at the info button contents.

I keep being impressed by what a good tool for teaching and learning a spreadsheet on a page of HTML can be. These info buttons you put in are an excellent feature.

I don't see any factual errors so I would advise going ahead and installing the upgrade. There may be, here and there, awkwardness in wording that you might decide to change later. But I wouldn't be a perfectionist in wording, you can always change a word or two later.

In the "proper recession speed" button you said "different to"when (at least in American English) it's more common to say "different from". But as long as that is what you meant to say, I would vote for keeping it as you have it. It's good for things to be in your own words, as you would normally say them. I have some other comments like that but they can wait---mostly cosmetic, nothing urgent.

It would be interesting to see if other people have comments.

 

[marcus]

Herewith my personal reactions. Don't have to take 'em too seriously, it is your creation and ultimately should be in your words. But I'll say what comes to mind. I don't recall ever hearing about "lookback distance". Clearer to just have info button about lookback time and not mention it. Might get beginners confused.

At some point it might be good to mention that the Hubble radius is the proper distance at which recession is exactly = c, and within which recession is < c.

A lot of people might not be quite sure what the words "inverse" or "reciprocal" mean. So it might be plainer to say that the Hubble time is simply "one over the Hubble growth rate".

IMO the phrase "in the appropriate units" does not add much to your button about 1/H. It is the same quantity regardless of what units it is expressed in. I hope somewhere in your website's teaching material you can find a way to prompt people to themselves simply paste this into google, without the quotes:
"70.4 km/s per Mpc"
and also this:
"1/(70.4 km/s per Mpc)"

speed divided by distance = number per unit time
so they will find that google gives H as a small number per second, which it in fact IS! Though equally well a different small number per million years.
The choice of "km/s per Mpc" is an historical accident: a complicated junky way of talking about a number per unit time

distance divided by speed = time
so "Megaparsec per km/s" is actually a unit of time--a rather ugly one--and
they will find that google gives 1/H as a length of time (which it chooses to express in years, but could equally accurately express in seconds or some other unit of time.)

Physical quantities are themselves (to make an obvious point) regardless of what units they happen to be expressed in.
So 1/H is fundamentally a time
and H is fundamentally one over a time, i.e.a number per unit time (which is typically how we express frequencies and fractional growth rates.)

 

[Jorrie]

Herewith my personal reactions. Don't have to take 'em too seriously, it is your creation and ultimately should be in your words.

My sincere thanks for your comments and suggestions. I have massaged the info text somewhat and now uploaded it under the old url, so the link in your signature should take you there. I stuck to the old url so that older links would take readers to the new version. The first old url was just .../cosmocalc.htm and it also links to the latest version. However, I am temporarily leaving the .../cosmocalc_2012.htm link pointing to the version that you have commented upon, for continuity.

As far as the relative merits of HTML and spreadsheets are concerned, I find spreadsheets hugely useful, but it is difficult to get numerical integration for large ranges with small increments to be 'user friendly'. HTML/Java script solves that problem, but then it is difficult to make large scrolling tables. (I'm probably just not good enough of a 'spread-sheeter' or Java-scriptwriter).

Question: is there a reason for your introduction of the symbol Y for 1/H? I haven't seen it anywhere else. Apart from it ringing a nice 'units-bell', are there other considerations?

 

[marcus]

I really like this version:
http://www.einsteins-theory-of-relativity-4engineers.com/beta_3/cosmocalc_code_2012.htm
so I changed over to it, in my signature, from the older one:
http://www.einsteins-theory-of-relativity-4engineers.com/cosmocalc_2010.htm

Visually the new one "beta 3" is handsome.
I like the blue info button texts (pedagogically they add a lot)
I really like the "time for a one percent increase" entry.

The URL for the new version says "beta 3", is it this "beta 3" version which will now become the official one? If so, I'm delighted.
If not, would you be willing to leave "beta 3" online at that address, so that I can keep that in my signature and work with that?
========================
I'm worried because the "beta 3" apparently didn't go in. When I click on the older link I don't get that one. BTW it's not vanity You didn't need to mention me in the acknowledgments, although it was kind. Quite apart from acknowledgments, I like the new version because it's got considerably more appeal and pedagogical value!
========================
You asked about the choice of Y in this thread, where I'm experimenting with different ways to present the Friedmann model to beginners. No special reason. Could be J and J_∞ instead of Y and Y_∞. Or Q, or W.

It would be cumbersome to write T_H or T_Hubble. Gum things up with subscripts. I just want a single letter for the Hubble time 1/H, and it could be any letter that is not too closely associated with something else. E or M or V or P would not be good. I just picked Y without thinking much about it. Let's see how the basic differential equation looks with different letters.

Y' = (3/2)[ 1 - (Y/Y_∞)²], where Y is the current value of the Hubble time 1/H.

J' = (3/2)[ 1 - (J/J_∞)²], where J is the current value of the Hubble time 1/H.

Q' = (3/2)[ 1 - (Q/Q_∞)²], where Q is the current value of the Hubble time 1/H.

Do you have any preference? or ideas about choice of symbol? The more I think about it the more I like Y. It reminds me of the letter T, the usual symbol for an interval of time, but with the arms raised a little. It's easy to remember that it stands for a really important longish interval of time (at least for me.)
========================

I like the "1% Hubble time" quantity we were talking about. A lot. It is the time needed for distances to grow by 1% (at their current growth speed.) That is a good handle on distance growth for beginners to get hold of.

Picture the world containing a lot of stationary observers (or objects) and a whole bunch of distances between them. Each distance is growing according to how big it is--in proportion to its current size. At a speed that is proportional to its current length.

I think that is how to picture expansion, and saying "1% Hubble time" to beginners inculcates that picture. It plants the right seed in their minds, when you say "time needed for distances to grow by 1%, at the present rate".

The picture planted is not of MOTION but of growth like in a bank account. It is so important to get that right, at the start.

So then we move on from there. Currently the 1%Y is 139 My which means 1/139% growth in a million years. And 1% of the eventual Y, denoted Y_∞, is 163 My which means 1/163% growth in a million years.
And the Y time itself is increasing according to this simple differential equation:
Y' = (3/2)[ 1 - (Y/Y_∞)²] which if we plug in current values 139 and 163 gives
Y' = (3/2)[ 1 - (139/163)²] = (3/2)[ 1 - 0.728] = 0.41

So we even have a handle on how rapidly the Hubble time has been growing recently and will be growing in the immediate future. Great! And nobody has, so far, had to look at a Megaparsec, or 8πG, or imagine galaxies zooming (thru space ) at km/s, or some kind of "dark energy" pushing on the galaxies so they will zoom faster thru space We are talking about an enormous avoidance of fallacious mental garbage and clutter, basically just by including this "1% Hubble time" idea and letting beginners get familiar with it. Have you to thank in part. Thanks

 

[Jorrie]

I'm worried because the "beta 3" apparently didn't go in. When I click on the link in my sig I don't get that one.

I think it may be just a cache issue on some server, because the one on your sig is actually the newest version, which is the beta 3 version with minor improvements in the text and the footnotes. If CTRL-reload does not help, you can stick this URL into your sig. It should force the server to load the newest version. The beta_3 one will eventually disappear, because it is a little confusing to have both versions, almost identical. I just left it there because it was commented on.

Edit: Ah, I see you have changed the sig. I would be more happy if you rather change it back, or to the newer URL that I gave above. Believe me, the wording is 'slightly better'. There is also an extra footnote.

You asked about the choice of Y in this thread, where I'm experimenting with different ways to present the Friedmann model to beginners. No special reason. Could be J and J_∞ instead of Y and Y_∞. Or Q, or W.

Different symbols were not my issue, but rather the not using 1/H. I think you have motivated it well, but whether if would help amateur cosmologists or hinder them, I'm not sure. There is such a huge legacy of equations with H in them...

 

[marcus]

Excellent! I pressed reload and the new one appeared. I will change link in sig to
http://www.einsteins-theory-of-relativity-4engineers.com/cosmocalc_2010.htm

Hubble radius corresponds to a redshift of 1.41
It is the distance of an object or observer which is NOW receding at speed c.
Your info bubble could advise the user to put in z=1.41 and see,
and the proper distance increase (as of present moment) will turn out to be 0.9999 c
which is nice and close to c, close enuff I'd say.

I think your info bubble should have the word "now" in this sentence, and come to a full stop.
"It is also the proper distance at which the proper recession speed 'now' equals the speed of light."

What follows this sentence could simply be erased.
==================

I should explain a bit. The Hubble radius is not the same as the CEH (cosmic event horizon). Even professionals might sometimes confuse them (Lineweaver and Davis "Expanding Confusion" article 2003 describes instances of such confusion).

If you look at this figure, at the top layer (where the scale is proper distance):
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure1.jpg
you will see the CEH plotted and within it the Hubble radius or "Hubble sphere" also plotted.

If you look at where the 'now' line crossed the figure you will see that from there on up the CEH curve is nearly vertical. That means that the CEH is already NEARLY at its limiting value of 16.3 billion ly proper distance.

But the Hubble radius still has a considerable ways to go. It is currently only 13.9 billion ly proper distance.
The area between the two curves is cross-hatched.

I don't know the exact current value of the CEH as of today. I only know that it is approaching 16.3 and is already so near that I can say "around 16". It might currently be 15.9, but why quibble? You can see that it's around 16 just by looking at the figure.

What we need for calculation, what plays a major role, is the *eventual* CEH of 16.3. That is also the limiting value of the Hubble radius. This is the number 163 which essentially has the information about Lambda, what the longterm expansion rate is going to be 10 billion years from now when distances are twice, and volumes are 8 times what they are today and all the matter and radiation density is 1/8 what it is today and the expansion rate is really only reflecting the value of Lambda.

H_∞ is 1/163 percent per million years. And the square of that rate is essentially Lambda (actually Λc²/3, Lambda up to some constants.) A residual spacetime curvature reflected in a longterm expansion rate. So that limiting CEH of 16.3 billion ly, contains really basic information.

My feeling is that it is too much to try to explain in the calculator's information bubble attached to the Hubble radius (of 13.9 billion ly). It is a separate topic. Beautiful and exciting but confusing if mixed in with the Hubble radius. Maybe there is some way to fit it in, in its own place, not sharing room with Hubble radius. Have to think about that later.

 

[marcus]

Since we've turned a page, I'll bring forward the earlier table, and add some new rows closer to the start of expansion. BTW the table shows the interesting then-distance maximum around 5.8 billion ly. To remind anyone who happens to be reading, the numbers in this table were gotten with the help of Jorrie's calculator. The calculator gives multidigit precision and I've rounded off. Hubble rates at various times in past are shown both in conventional units (km/s per Mpc) and as fractional growth rates per d=10⁸y. The first few columns show lookback time in billions of years, and how the Hubble rate has been declining, while the Hubble radius (reciprocally) has extended out farther. The columns on the right show the proper distance (in Gly) of an object seen at given redshift z both now and back when it emitted the light we are currently receiving. The numbers in parenthesis are fractions or multiples of the speed of light showing how rapidly the particular distance was growing.

Standard model with WMAP parameters 70.4 km/s per Mpc and 0.728. 
Lookback times shown in Gy. Hubble growth rate H and time Y=1/H shown
using time unit d = 10[SUP]8[/SUP] y. The "now" and "then" distances are shown in Gly,
with their growth speeds in c.
time      z     H(conv)   H(d[SUP]-1[/SUP])   Y=1/H(d)     now         then 
   0     0.000     70.4   1/139    139      0.0          0.0
   1     0.076     72.7   1/134    134      1.0(0.075)   1.0(0.072)
   2     0.161     75.6   1/129    129      2.2(0.16)    1.9(0.14)
   3     0.256     79.2   1/123    123      3.4(0.24)    2.7(0.22)
   4     0.365     83.9   1/117    117      4.7(0.34)    3.4(0.29)          
   5     0.492     89.9   1/109    109      6.1(0.44     4.1(0.38
   6     0.642     97.9   1/100    100      7.7(0.55)    4.7(0.47)
   7     0.824    108.6   1/90      90      9.4(0.68)    5.2(0.57)
   8     1.054    123.7   1/79      79     11.3(0.82)    5.5(0.70)
   9     1.355    145.7   1/67      67     13.5(0.97)    5.7(0.86)
  10     1.778    180.4   1/54      54     16.1(1.16)    5.8(1.07)
  11     2.436    241.5   1/40      40     19.2(1.38)    5.6(1.38)
  12     3.659    374.3   1/26      26     23.1(1.67)    5.0(1.90)
 13.0    7.170    860.5  1/11.36  11.36    29.2(2.10)    3.6(3.15)
 13.1    7.979    991.0  1/9.87    9.87    30.0(2.16)    3.3(3.38)
 13.2    9.021   1168.0  1/8.37    8.37    30.9(2.23)    3.1(3.69)
 13.3   10.432   1422.9  1/6.87    6.87    32.0(2.30)    2.8(4.07)
 13.4   12.469   1819.5  1/5.37    5.37    33.2(2.39)    2.5(4.59)
 13.5   15.754   2524.9  1/3.87    3.87    34.7(2.50)    2.1(5.35)
 13.6   22.221   4123.1  1/2.37    2.37    36.7(2.64)    1.6(6.66) 
 13.7   44.320  11277.6  1/0.87    0.87    39.8(2.87)    0.9(10.13)

Abbreviations used in the table:
"time" : Lookback time, how long ago, or how long the light has been traveling.
z : fractional amount distances and wavelengths have increased while light was in transit. Arriving wavelength is 1+z times original.
H : Hubble expansion rate, at present or at times in past. Distances between observers at rest grow at this fractional rate--a certain fraction or percent of their length per unit time.
H(conv) : conventional notation in km/s per Megaparsec.
H(d^-1) : fractional increase per convenient unit of time d = 10⁸ years.
Y=1/H: Hubble time. 1% of the current Hubble time is how long it takes distances to increase by 1%, growing at the current rate. The current value of Y is 139 d = 13.9 billion years.
Hubble time is proportional to the radius = c/H: distances smaller than this grow slower than the speed of light. Current Hubble radius is of course 13.9 billion ly (proper distance)
"now" : distance to object at present moment of universe time (time as measured by observers at CMB rest). Proper distance i.e. as if one could freeze geometric expansion at the given moment.
"then" : distance to object at the time when it emitted the light.

Remember that "proper" distance, the distance used in Hubble law to describe expansion, is "freezeframe". The proper distance at a given moment in Universe time is what you would measure (by radar or string or whatever usual method) if at that moment you could stop the expansion process long enough to make the measurement.
The Hubble law describes the expansion of distances between observers at rest with respect to the background of ancient light and the process itself. Observers who see the ancient light and the expansion process approximately the same in all directions, e.g. no Doppler hotspots.
The field of an observer's view is not conical, but rather it is pear-shape because distances were shorter back then. Here is a picture of an Anjou pear.
http://carrotsareorange.com/wp-content/uploads/2010/05/pear-anjou.jpg
Here is Lineweaver's spacetime diagram:
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure1.jpg
The upperstory figure, with horizontal scale in proper distance, shows the lightpear outline.
Here is Lineweavers plot of the growth of the scalefactor R(t), which models the growth of all distances between observers at universe-rest (at rest with respect to background.)
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure14.jpg
The dark solid line is according to standard model parameters. Various other cases are shown as well.

 

[marcus]

Y-prime cosmo intro

The conventional formulation of cosmology is based on the Friedmann equation governing the fractional distance growth rate H. For simplicity we assume spatial flatness and don't try to cover the brief radiation-dominated era at the outset of expansion.

Simplification results if we substitute the variable Y = 1/H in the Friedman equation and let it tell us the evolution of the Hubble time Y instead of the Hubble growth rate.

The Friedmann equation becomes:
Y' = (3/2)[1 - (Y/Y_∞)²]
where Y_∞ = 16.3 billion years, the longterm limit of the Hubble time. Just to have a name for it, I will call this the "Y-prime equation".

This equation is dimensionless (change in time per unit time) which is convenient--we don't have to worry about units. Since the current Y = 13.9 billion years, the current value of Y' = 0.41.

In the early universe Y is small, so the squared term is suppressed. That means that the slope of the Y curve is essentially 1.5. The slope goes to zero as Y → Y_∞.

Once one knows Y as a function of time, the evolution of other quantities can be calculated: redshift at various lookback times, the evolution over time of the scalefactor and the matter density. So this gives an elementary way to approach cosmology. Calculating these other quantities from Y requires other physical contants such as G and c.

However the only constant in the Y' equation is Y_∞, which is basically the cosmological constant in a different guise. Lambda = 3(1/cY_∞)²

If you look at the table in the preceding post you will see that over the lookback time interval 13.0 to 13.7, the Hubble time Y is growing with slope 1.5. As you go up the column, each successive number is 1.5 larger. After time 13.0 the slope gradually diminishes towards the present Y' value around 0.4.

 

[Jorrie]

I think your info bubble should have the word "now" in this sentence, and come to a full stop.
"It is also the proper distance at which the proper recession speed 'now' equals the speed of light."

What follows this sentence could simply be erased.
==================

Thanks Marcus, I messed up there in my eagerness to get the 'info bubbles' out.
I will correct the erroneous text...

Will also think about the event horizon and H_inf and how to present it somewhere.

 

[marcus]

No problem! You did a splendid job in short order. The calculator is a valuable resource. I just used it to expand and improve the table some. It is remarkable how the equation for Y' predicts a slope of 3/2 for the Hubble time curve---and that slope of 1.5 shows up in the table over the interval 13.7 to 13.0 lookback time. Because where Y is small the square term is suppressed by the Y_∞ constant in the denominator. I like the simple way the Hubble time (Y) increases with time.

 

[Mark M]

Marcus, considering the number of great posts here, there's a bit of a problem - since there are 24 pages, much of the information is difficult to find. So, I thought I'd put some of your posts together into one page. I took posts starting from around page 18, where you started doing calculations.

Any writing in bold is added by me, to try and add some pieces of information that may be helpful. Here it is:

View attachment cosmo.pdf

It isn't nearly finished, I'm still going to

1)Get the equations into Latex

2)Make the chart into a proper chart using Office.

3)Since you were trying to keep readers updates as new pages turned, you repeated some calculations. So, many of the posts contain similar calculations, I intend to make a few edits so that some things won't appear many times, over and over.

4)And of course, add more information as you post it.

Let me know if there's anything else you'd like changed/added. And of course, 4) is the reason it's currently so long.

 

[marcus]

Marcus, considering the number of great posts here, there's a bit of a problem - since there are 24 pages, much of the information is difficult to find. So, I thought I'd put some of your posts together into one page. I took posts starting from around page 18, where you started doing calculations.

Any writing in bold is added by me, to try and add some pieces of information that may be helpful. Here it is:

View attachment 49886

It isn't nearly finished, I'm still going to

1)Get the equations into Latex

2)Make the chart into a proper chart using Office.

3)Since you were trying to keep readers updates as new pages turned, you repeated some calculations. So, many of the posts contain similar calculations, I intend to make a few edits so that some things won't appear many times, over and over.

4)And of course, add more information as you post it.

Let me know if there's anything else you'd like changed/added. And of course, 4) is the reason it's currently so long.

Sounds like an excellent plan, Mark. You might be able to construct a concise tutorial on basic cosmology using this approach. Feel free to use material from this thread, and also to adapt the approach as you see fit.

It would be a big help if, when you put equations into Latex, you make the Latex source available so that I can copy it and use it in my posts here. You could start your own thread here at Cosmo forum, or make the Latex available some other way. If you want to do that, I would appreciate it.

The main equation of this approach is the Y-prime equation.
Y' = [3/2](1 - [Y/Y_∞]²)
It would be prettier in Latex and it would be nice to have a Latex version. Using this equation one can numerically reconstruct the history of the Hubble time Y(t) going back in time to a lookback time of around 13.7 billion years.
Only two inputs are needed for this: Y_now=13.9 Gy and Y_∞=16.3 Gy. From that the Y(t) history can be generated from that differential equatiion and then other stuff about the history of the U can be based on the Y(t) history.

The other equation it would be especially nice to have in Latex is the one giving the matter density ρ at any time in the past, once you know Y.

ρ = 3[1/Y² - 1/Y_∞²]/(8πG)

If you want to make a Latex source for that, of course arrange it as seems best, you might want to put 3/(8πG) at the beginning. That might look better in Latex.

My pedagogical leaning is always to, as much as possible, get beginners to use the google calculator and Jorrie's calculator (which I think is currently the best online cosmo calculator) to calculate stuff on their own. I think active hands-on cosmology is better than mere book cosmology.

You can see how, once a person has the Y(t) history for lookback times as far as t=13.7 Gy, they can calculate the matter density ρ(t) also for lookback times as far back as t=13.7 Gy.

Beyond that, going from lookback 13.700 to 13.757 where the model starts, is going too far because it gets into the radiation era. Radiation density behaves differently in expansion, from matter density. So equations have to be modified and I want to keep it simple.

You can also see how once a person knows how to calculate Y(t) and ρ(t) for some time t in the past, they can calculate z(t), the corresponding redshift.

(1+z)³ = ρ(then)/ρ(now)

So those 3 equations. The Y-prime, the rho, and the zee equation would be nice to have in Latex. They suffice to let you reconstruct the history of the cosmos back to lookback time t=13.7 Gy and they are, I think, very basic and easy to use.

You should feel free to use that "lesson plan" if you like it, or construct your own ordering of the material. AFAICS there is no reason to have just ONE tutorial. I might get around to editing and constructing one (using your Latex if you make it available). I take a kind of experimental approach. You learn by experimenting with different attempts at organization presentation explanation.

Be advised that this is "quick and dirty FLAT cosmology". The spatial curvature term in the Friedmann equation is a pain in the butt. If I was teaching Freshmen I would simply omit it. The U is obviously so near spatially flat that it might as well be treated that way, and in fact the professionals do treat it that way most of the time.
The issue hasn't been settled. Planck mission data (next year?) will probably narrow the curvature confidence interval down some more.

 

[Mark M]

Okay, I'll make sure to add all of that information. Yes, I will definitely put all of the Latex into a file. By the way, if you're interested, there is a very good text to Latex converter where you use the above buttons and plain text and it converts it to Latex:

http://www.codecogs.com/latex/eqneditor.php

EDIT: Oh, and the Latex for the three equations above are:

Y' = [3/2](1-[Y/Y_∞]^2) becomes

$$Y'= \frac{3}{2} {\left(1- {\left(\frac {Y}{Y_∞} \right)}^2 \right)}$$

$$Y'= \frac{3}{2} {\left(1- {\left(\frac {Y}{Y_∞} \right)}^2 \right)}$$

ρ = 3[1/Y2 - 1/Y∞2]/(8πG) becomes

$$\rho = \frac {3{\left(\frac{1}{Y^2} -\frac {1}{Y_\infty ^2}\right )}} {8\pi G}$$

$$\rho = \frac {3{\left(\frac{1}{Y^2} -\frac {1}{Y_\infty ^2}\right )}} {8\pi G}$$

Finally, (1+z)3 = ρ(then)/ρ(now) becomes

$$(1+z)^3 = \frac {\rho  (then)}{\rho (now)}$$

$$(1+z)^3 = \frac {\rho (then)}{\rho (now)}$$

 

[marcus]

$${\left(\frac {Y_{now}}{Y_∞} \right)}^2 = 0.728$$

$$Y'= \frac{3}{2} {\left(1- {\left(\frac {Y}{Y_∞} \right)}^2 \right)}$$

$$\rho = \frac {3}{8\pi G}{\left(\frac{1}{Y^2} -\frac {1}{Y_\infty ^2 }\right)}$$

$$(1+z)^3 = \frac {\rho (then)}{\rho (now)}$$

$$(1+z)^3 = \frac {0.728}{1-0.728}{\left({\left(\frac {Y_∞}{Y} \right)}^2 -1\right)}$$

 

[Mark M]

$$\rho = \frac {3}{8\pi G}(\frac{1}{Y^2} -\frac {1}{Y_\infty ^2)$$

Oh, sorry about that. Fixed:

$$\rho = \frac {3}{8\pi G}{\left(\frac{1}{Y^2} -\frac {1}{Y_\infty ^2}\right )}$$

$$\rho = \frac {3}{8\pi G}{\left(\frac{1}{Y^2} -\frac {1}{Y_\infty ^2}\right )}$$

 

[marcus]

Thanks. I updated the table in post#370
It's nice how the Y entries follow the differential equation between 13.0 and 13.7 billion years, going in steps of 100 million years at a time.
The Y-prime equation says that for small Y the slope of Y should be 3/2 = 1.5
so the increment at each step should be 1.5. Which it is!
Then as Y gets larger the slope should decline
which it does.
Between 10.0 and 13.0 where you go in steps of 1 billion, the steps are ten times as large so you so expect increments of 15 at first (that is 1.5 x 10) but declining.
So it turns out the table illustrates the differential equation.
Especially in the small steps part of the table---13.0-13.7.

 

[Mark M]

Thanks. I updated the table in post#370
It's nice how the Y entries follow the differential equation between 13.0 and 13.7 billion years, going in steps of 100 million years at a time.
The Y-prime equation says that for small Y the slope of Y should be 3/2 = 1.5
so the increment at each step should be 1.5. Which it is!
Then as Y gets larger the slope should decline
which it does.
Between 10.0 and 13.0 where you go in steps of 1 billion, the steps are ten times as large so you so expect increments of 15 at first (that is 1.5 x 10) but declining.
So it turns out the table illustrates the differential equation.
Especially in the small steps part of the table---13.0-13.7.

Oh, I used the chart you posted earlier by accident. I'll fix that.

 

[Mark M]

Okay, here's the updated chart:

I'll delete the other one.

 

[marcus]

Great! looking good, thanks. One minor thing: the fifth column is not a distance but is rather a time, expressed in convenient units d=10^8 years.

I would not say "Hubble radius" for the heading. I would say "Hubble time (d)"

Since there are only two columns that give distance, it would be more concise to erase the legend "all distances in billions of light years" and simply add the symbol (Gly) in the two relevant headings.

So the headings go:
Lookback time (Gyr)
z(redshift)
H(conv.)
H(per d)
Hubble time(d)
Distance now (Gly)
Distance then (Gly)

If you don't think it would be too cluttered, the conventional H heading could, instead of H(conv.), say
H(km/s per Mpc)
because the conventional unit for H is km/s per Mpc
and the parentheses in the headings are mainly to show what unit is used in that column.

One thing that occurs to me, under the main headline it could say something like
"Distances now and at lookback time are shown with their growth speeds (in c)"

that would give a clue to the reader that the numbers in parens, right after the distance number, are how fast that distance is growing, expressed in c units.

 

[Mark M]

Great! looking good, thanks. One minor thing: the fifth column is not a distance but is rather a time, expressed in convenient units d=10^8 years.

I would not say "Hubble radius" for the heading. I would say "Hubble time (d)"

Since there are only two columns that give distance, it would be more concise to erase the legend "all distances in billions of light years" and simply add the symbol (Gly) in the two relevant headings.

So the headings go:
Lookback time (Gyr)
z(redshift)
H(conv.)
H(per d)
Hubble time(d)
Distance now (Gly)
Distance then (Gly)

If you don't think it would be too cluttered, the conventional H heading could, instead of H(conv.), say
H(km/s per Mpc)
because the conventional unit for H is km/s per Mpc
and the parentheses in the headings are mainly to show what unit is used in that column.

Sure! I'll change that soon, when I get the chance. Using that for H(conv.) probably won't fit, is it fine if I put an asterisk next to it at the top of the column, and then write at the bottom of the page '*H(km/s per Megaparsec)'? I'll also add a note about Y being measured in units of time.

 

[marcus]

Ultimately, anything you do is good. Your own taste about what looks right, and judgement about what is clear gets the final say. But I'll make suggestions.
You have two lines. How about this for headings?

H
(km/s per Mpc)

H
(per d)

Hubble
time (d)

Distance
now (Gly)

Distance
then (Gly)

Also an idea I had a couple of posts back:

Under the title headline it could say something like
"Distances now and at lookback time are shown with their growth speeds (in c)"

 

[marcus]

Here's the current skeleton plan for a cosmo tutorial. Thanks to Jorrie and Mark for help and interest!

First of all, to get started this is where the main published parameter we need comes in, namely 0.728, the estimated square of the ratio of now Hubble time to eventual Hubble time. Y→Y_∞.

$${\left(\frac {Y_{now}}{Y_∞} \right)}^2 = 0.728$$

In terms of the convenient time unit d=10⁸ years, Y_now = 139 d
and Y_∞ = 163 d.

Given this data we can plot the curve Y(t) of Hubble time all the way back to a lookback time of 13.7 Gy. The standard model age is 13.757 Gy, so we come within 57 million years of start of expansion, close enough for an introductory treatment.

$$Y'= \frac{3}{2} {\left(1- {\left(\frac {Y}{Y_∞} \right)}^2 \right)}$$

Once we have Y(t) for lookback times from 13.7 Gy to present, we can calculate the redshift corresponding to any given lookback time (i.e. how far in the past the light we are now receiving was emitted.)

$$(1+z)^3 = \frac {0.728}{1-0.728}{\left({\left(\frac {Y_∞}{Y} \right)}^2 -1\right)}$$

EDIT: On good advice I think this should be changed to
$$(1+z)^3 = {\left({\left(\frac {Y_∞}{Y} \right)}^2 -1\right)}/ {\left({\left(\frac {Y_∞}{Y_{now}} \right)}^2 -1\right)}$$

The current estimated value of the denominator in this expression is 0.3736 = 1/0.728 - 1.

Various other things can also be calculated, knowing the Hubble time Y(t) in the past, for example the matter density (dark, ordinary, plus a minor contribution from radiation)

$$\rho = \frac {3}{8\pi G}{\left(\frac{1}{Y^2} -\frac {1}{Y_\infty ^2 }\right)}$$

As a check, the calculated matter density at some time in the past should agree with the redshift of light from a given moment in the past.

$$(1+z)^3 = \frac {\rho (then)}{\rho (now)}$$

===========================
I almost forgot to say! What is the Hubble time? Basically it gives a handle on the rate that distances are expanding at present or at some moment in the past.
One percent of the Hubble time Y(t) is the time it would take for a distance (at its current rate) to grow by one percent.

 

[Mark M]

Sounds good!

One problem that people often have when first learning about cosmology is getting the idea that galaxies are actually flying away from each other. Of course, what's really going on is that distances are growing. So, I wrote this (rather long) introduction to the idea of what it means to say the universe is expanding. If you're interested, here it is:

View attachment Cosmology.pdf

 

[marcus]

Sounds good!

One problem that people often have when first learning about cosmology is getting the idea that galaxies are actually flying away from each other. Of course, what's really going on is that distances are growing. So, I wrote this (rather long) introduction to the idea of what it means to say the universe is expanding. If you're interested, here it is:

View attachment 49908

Nice easy-to-read style, with clear illustrations. I think it would communicate to high school students, maybe even bright interested junior high. Avoids over-technical language.

The table is work in progress. You are working on one version. And I'm still trying different headings. right now it may be too dense with information. but I'm still trying to keep it all and not chop anything out. It could be nearing a final version. I look forward to seeing your next draft. Here is what I have at present:

The first few columns show lookback time in billions of years, and how the Hubble rate has been declining, while the Hubble time (reciprocally) has increased. The columns on the right show the proper distance (in Gly) of an object seen at given redshift z ---both now and back when it emitted the light we are currently receiving. The numbers in parenthesis are multiples of the speed of light showing how rapidly the particular distance was growing.
The tabulation is based on the standard cosmic model with parameters 70.4 km/s per Mpc and 0.728, corresponding to an expansion age of 13.757 billion years. Our reconstruction of expansion history only goes back to an estimated 13.700 billion years, so to within 57 million years of the start.

Expansion history.
Lookback times shown in Gy. Hubble growth rate H and Hubble time Y=1/H are shown
using time unit d = 10[SUP]8[/SUP] y. The "now" and "then" distances are shown in Gly,
with their growth speeds in c.
Lookback   z     H(conv)   H     Hubble time  now         then 
time(Gy) redshift        (per d)   (d)       (Gly)       (Gly)
   0     0.000     70.4   1/139    139      0.0          0.0
   1     0.076     72.7   1/134    134      1.0(0.075)   1.0(0.072)
   2     0.161     75.6   1/129    129      2.2(0.16)    1.9(0.14)
   3     0.256     79.2   1/123    123      3.4(0.24)    2.7(0.22)
   4     0.365     83.9   1/117    117      4.7(0.34)    3.4(0.29)          
   5     0.492     89.9   1/109    109      6.1(0.44     4.1(0.38
   6     0.642     97.9   1/100    100      7.7(0.55)    4.7(0.47)
   7     0.824    108.6   1/90      90      9.4(0.68)    5.2(0.57)
   8     1.054    123.7   1/79      79     11.3(0.82)    5.5(0.70)
   9     1.355    145.7   1/67      67     13.5(0.97)    5.7(0.86)
  10     1.778    180.4   1/54      54     16.1(1.16)    5.8(1.07)
  11     2.436    241.5   1/40      40     19.2(1.38)    5.6(1.38)
  12     3.659    374.3   1/26      26     23.1(1.67)    5.0(1.90)
 13.0    7.170    860.5  1/11.36  11.36    29.2(2.10)    3.6(3.15)
 13.1    7.979    991.0  1/9.87    9.87    30.0(2.16)    3.3(3.38)
 13.2    9.021   1168.0  1/8.37    8.37    30.9(2.23)    3.1(3.69)
 13.3   10.432   1422.9  1/6.87    6.87    32.0(2.30)    2.8(4.07)
 13.4   12.469   1819.5  1/5.37    5.37    33.2(2.39)    2.5(4.59)
 13.5   15.754   2524.9  1/3.87    3.87    34.7(2.50)    2.1(5.35)
 13.6   22.221   4123.1  1/2.37    2.37    36.7(2.64)    1.6(6.66) 
 13.7   44.320  11277.6  1/0.87    0.87    39.8(2.87)    0.9(10.13)

Terms and abbreviations used in the table:
Lookback time: how long ago, or how long the light has been traveling.
z : fractional amount distances and wavelengths have increased while light was in transit. Arriving wavelength is 1+z times original.
H : Hubble expansion rate, at present or at times in past. Distances between observers at rest grow at this fractional rate--a certain fraction or percent of their length per unit time.
H(conv) : conventional notation in km/s per Megaparsec.
H(per d) : fractional increase per convenient unit of time d = 10⁸ years.
Y=1/H: Hubble time. 1% of the current Hubble time is how long it takes distances to increase by 1%, growing at the current rate. The current value of Y is 139 d = 13.9 billion years.
Hubble time is proportional to the Hubble radius = c/H: distances smaller than this grow slower than the speed of light. Current Hubble radius is 13.9 billion ly (proper distance)
"now" : distance to object at present moment of universe time (time as measured by observers at CMB rest). Proper distance i.e. as if one could freeze geometric expansion at the given moment.
"then" : distance to object at the time when it emitted the light.

The Hubble law describes the expansion of distances between observers at rest with respect to the background of ancient light and the process itself: Observers who see the ancient light and the expansion process approximately the same in all directions, e.g. no Doppler hotspots.
The field of an observer's view can be thought of as pear-shape because distances were shorter back then. Here is a picture of an Anjou pear.
http://carrotsareorange.com/wp-content/uploads/2010/05/pear-anjou.jpg
Here is Lineweaver's spacetime diagram:
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure1.jpg
The upperstory figure, with horizontal scale in proper distance, shows the lightpear outline.
Here is Lineweavers plot of the growth of the scalefactor R(t), which models the growth of all distances between observers at universe-rest (at rest with respect to background.)
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure14.jpg
The dark solid line is according to standard model parameters. Various other cases are shown as well.

 

[Jorrie]

$${\left(\frac {Y_{now}}{Y_∞} \right)}^2 = 0.728$$

Marcus, the simplified equations have a nice look and feel, but I guess it may be better to leave the cosmological constant in its \Omega_\Lambda form, because the best-fit value (0.728) is bound to change at some time, as more accurate data come in.

I'm still trying to figure out whether the usage of Y rater than 1/H has any benefit. Equations like
(1+z)^3 = \frac {\Omega_\Lambda}{1-\Omega_\Lambda}{\left({\left(\frac {H}{H_∞} \right)}^2 -1\right)}
are as clear and have the benefit of standard parameter names.

Maybe <br /> Y&#039;= \frac{3}{2} {\left(1- {\left(\frac {Y}{Y_∞} \right)}^2 \right)}
would have looked more awkward in 1/H form. (?)

 

[marcus]

Marcus, the simplified equations have a nice look and feel, but I guess it may be better to leave the cosmological constant in its \Omega_\Lambda form, because the best-fit value (0.728) is bound to change at some time, as more accurate data come in...

Omega_Lambda is not the cosmological constant. It only has meaning in relation to the current value of H. Very few people seem to know the value of the cosmological constant Lambda (as a reciprocal area, or scalar curvature.

I think the easiest way to explain, from scratch, what the number 0.728 actually means is to say that

0.728 = (Y_now/Y_∞)²

or if you prefer the H variable (the reciprocal 1/Y) then what the number means is

0.728 = (H_∞/H_now)²

In other words, what 0.728 actually tells you is how big H_∞ is compared with H_now.

0.728 tells you what the long-term growth rate is, compared to the present growth rate.

It is precisely what tells me that if today's growth rate is 1/139 then the eventual growth rate is 1/163. Because their ratio is sqrt 0.728.
You know the saying you don't really understand something unless you can explain it to your grandmother, or some other friendly interested layperson with zero background.

You are welcome to think your way of explaining that datum to granny is better! First tell about critical density rho crit and give the formula for rho crit and say what it is in joules per cubic meter or grams etc and then talk about fractions of rho crit and then start talking BS about some fictional "energy" with special properties, and by then granny does not have a clue what youre saying.

If I had to explain that datum I would say what I just bolded. Granny can understand the interest rate at the bank and that the rate might decline over time to some slightly lower growth rate.

the ratio of present and future growth rates, simple enough idea.

So I choose to go that way. We'll see.

 

[marcus]

Jorrie, your objection seems to be that the datum 0.728 might change sometime.
Would you say this places our approach at a disadvantage?
I don't think it does, actually Suppose that the Planck mission and other observations in progress comes up with a new value of (H_∞/H_now)².

Mind you, that's basically what they are measuring---the evolution of the growth rate over time. They aren't measuring some fictitious "energy"---they are tracking growth rate and fitting a curve, something like a Y(t) curve or a H(t) curve.

So suppose they revise the estimate of (H_∞/H_now)² and the new estimate is 0.736 instead of 0.728.
For simplicity say H_now is still 1/139
So? We just revise H_∞ according to the new estimate of the ratio!

139/sqrt 0.736 = 162.0, so call it 162. The new H_∞ = 1/162 and the new Y_∞ = 162

 

[Jorrie]

In other words, what 0.728 actually tells you is how big _now is compared with H_now.

0.728 tells you what the long-term growth rate is, compared to the present growth rate.

I agree with what you said, but sticking in a numerical value like 0.728 is not so good; I guess you should then give it another symbol and say that it is determined experimentally and is bound to change. It is essentially \Omega_{\Lambda(now)} and also changes over cosmic time, like H.

I understand your drive based on a fundamental constant like \lambda and get away from 'energy density of the vacuum', which is a slippery concept to explain to grandmother.

Maybe give the ratio (H_∞/H_{now})^2 a new symbol?

 

[marcus]

I agree with what you said, but sticking in a numerical value like 0.728 is not so good; I guess you should then give it another symbol and say that it is determined experimentally and is bound to change. It is essentially \Omega_{\Lambda(now)} and also changes over cosmic time, like H.

I understand your drive based on a fundamental constant like \lambda and get away from 'energy density of the vacuum', which is a slippery concept to explain to grandmother.

Maybe give the ratio (H_∞/H_{now})^2 a new symbol?

I understand your point now! Duh! Of course. We can't use the raw number we need a symbolic expression for it. Exactly for the reason you said. It might change.
Why didn't I see what you were driving at immediately? OK I'll think about that and try to come up with something.

The normal thing to do is grab a Greek letter, but let's see how this looks

Y_now:∞= Y_now/Y_∞
Y_now/∞= Y_now/Y_∞
and in cosmology the present value is tagged with a subscript zero, like H₀ is standard notation for H_now
So how about
Y_o∞= Y_o/Y_∞
Y_o:∞= Y_o/Y_∞
Y_o/∞= Y_o/Y_∞
Or how about the letter ξ which I don't recall seeing much of lately?
ξ = Y_o/Y_∞

At the moment I'm leaning towards this way of handling it:
$$(1+z)^3 = {\left({\left(\frac {Y_∞}{Y} \right)}^2 -1\right)}/ {\left({\left(\frac {Y_∞}{Y_{now}} \right)}^2 -1\right)}$$
The current estimated value of the denominator in this expression is 0.3736 = 1/0.728 - 1.
So for practical purposes, until the estimate 0.728 changes, all we do is divide by 0.3736.
But here's another idea:
$$(1+z(t))^3 = (Y_{∞/t}^2 -1)/(Y_{∞/now}^2 -1)$$

That is, create symbols for the ratios of Hubble times. Several equations only depend on the times through their ratios, so having symbols for the ratios of the times would make the equations more compact. How would it go down with second year undergraduates, I wonder?

 

[Jorrie]

At the moment I'm leaning towards this way of handling it:
$$(1+z)^3 = {\left({\left(\frac {Y_∞}{Y} \right)}^2 -1\right)}/ {\left({\left(\frac {Y_∞}{Y_{now}} \right)}^2 -1\right)}$$
The current estimated value of the denominator in this expression is 0.3736 = 1/0.728 - 1.
So for practical purposes, until the estimate 0.728 changes, all we do is divide by 0.3736.
But here's another idea:
$$(1+z(t))^3 = (Y_{∞/t}^2 -1)/(Y_{∞/now}^2 -1)$$

That is, create symbols for the ratios of Hubble times. Several equations only depend on the times through their ratios, so having symbols for the ratios of the times would make the equations more compact. How would it go down with second year undergraduates, I wonder?

It starts to look all the more promising to me. I would prefer the latter of the two, because when you define Y, you can just as well define all the different subscripts and combinations. The fact that it is the ratio of two Hubble constants for different times, makes it appealing to me.

I also think 2nd-years will appreciate the more compact and easier to remember/recognize (and write) equation. It may be more of a problem for established cosmologists, where too much simplification/normalization becomes a hindrance. But since that's not your usage, it should be fine.

 

[marcus]

It's encouraging that you see the notation as moving in the right direction. I want to get back to something you said earlier (my original response was out of order) about the Y'-prime equation.

...Maybe <br /> Y&#039;= \frac{3}{2} {\left(1- {\left(\frac {Y}{Y_∞} \right)}^2 \right)}
would have looked more awkward in 1/H form. (?)

Right. I think you spotted the key factor. Try writing a differential equation for H. It gets messy. Constants like G and 8π and c come in, units come in. H' is dimensionful. The units are 1 over time squared. The density may get into the picture. So you have more things to think about.

By contrast, the Y' equation needs only the one constant, and the equation is dimensionless. Y' is simply a number. the units cancel out. change in time, per time.
In the fairly early universe Y' hovers near 1.5 for the greater part of a billion years and then drifts down to present value around 0.4.
The units play no role. Saying 0.4 years per year is the same as saying 0.4 century per century.

That equation does the work of the conventional Friedmann and conventional Raychaudhuri equations, in generating a curve from which you can calculate the other parts of the expansion history. and it is simpler than either conventional equation.

That is why I'm thinking that it might be a good basis for a beginner's tutorial.
For ADVANCED students it's a different matter. they can handle messier equations, spatial curvature term, radiation era etc etc. So of course let them wrestle with the complete unsmplified animal.

Since we're experimenting with the Y_now/∞ notation for ratios of Hubble times, let's try using that ratio symbol in the basic equation.

<br /> Y&#039;(t) = \frac{3}{2} {\left(1- {\left(\frac {Y(t)}{Y_∞} \right)}^2 \right)}

<br /> Y&#039;(t) = \frac{3}{2} {\left(1- Y_{t/∞}^2 \right)}

If that seems too compressed, one can always temporarily expand it back to the version on previous line, to explain something. Your point about compactness (at end of post #393) is well taken.

 

[George Jones]

<br /> Y&#039;(t) = \frac{3}{2} {\left(1- {\left(\frac {Y(t)}{Y_∞} \right)}^2 \right)}

<br /> Y&#039;(t) = \frac{3}{2} {\left(1- Y_{t/∞}^2 \right)}

I haven't been following this thread, so I might have missed some things. Did anyone write down the easy analytical solution to this separable differential equation?

<br /> \int^{Y\left(t\right)}_{Y\left(0\right)} \frac{dY}{1 - \left( \frac{Y}{Y_\infty} \right)^2} = \frac{3}{2} \int^t_0 dt&#039;<br />
gives

<br /> Y\left(t\right) = Y_\infty \tanh\left( \frac{3t}{2Y_\infty} \right)<br />
for Y\left(0\right) = 0.

 

[marcus]

Beautiful! Thanks, George! Now we are cooking

Following the discussion we just had, I'll emend the skeleton outline for a cosmo tutorial I sketched out in post#385.

We want to understand the expansion history and if possible reproduce it with some simple equations. The Hubble time Y(t) gives a handle on the rate distances are expanding at present or at some moment in the past.
One percent of Y(t) is how long it would take for a distance (at its current rate) to grow by one percent.

Two important ones to know (expressed in billions of years, abbr. Gy) are Y_now = 13.9 Gy
and the longterm future value that Y(t) is tending towards
Y_∞ = 16.3 Gy.

Every so often it's convenient to have a slightly smaller unit of time, a tenth of a billion, so we'll occasionally use d=10⁸ years, as an auxilliary unit. In those terms:
Y_now = 139 d
Y_∞ = 163 d.
One percent of Y_now = 139 million years. So, recalling what was just said, that is how long it would take a given cosmological distance to grow by one percent (at its current rate.) Alternatively one can think of distances currently growing 1/139 of a percent per million years.

A couple of equations we use depend on RATIOS of Hubble times, so as a compact notation we will write:
$$Y_{now/∞} = \frac {Y_{now}}{Y_∞} $$ The square of this ratio is currently estimated by cosmologists to be about 0.728. It's a key number and may change slightly in future, with increasingly accurate observations. For the time being we write:
$$Y_{now/∞}^2 = 0.728$$ Given these data we can plot the curve Y(t) of Hubble time all the way back to a lookback time of 13.7 Gy. The standard model age is 13.757 Gy, so we come within 57 million years of start of expansion, close enough for an introductory treatment. Lookback time is a negative or "backward" timescale and a bit awkward at first--it takes a little getting used to, but one copes. Often the minus sign is omitted.
Y&#039;(t) = \frac{3}{2} {\left(1- {\left(\frac {Y(t)}{Y_∞} \right)}^2 \right)}
Y&#039;(t) = \frac{3}{2} {\left(1- Y_{t/∞}^2 \right)}
Once we have Y(t) for lookback times from 13.7 Gy to present, we can calculate the redshift z(t) corresponding to any given lookback time t (i.e. how far in the past the light we are now receiving was emitted.)
$$(1+z(t))^3 = {\left({\left(\frac {Y_∞}{Y(t)} \right)}^2 -1\right)}/ {\left({\left(\frac {Y_∞}{Y_{now}} \right)}^2 -1\right)}$$
$$(1+z(t))^3 = {\left(Y_{∞/t}^2 -1\right)}/ {\left(Y_{∞/now}^2 -1\right)}$$
The current estimated value of the denominator in this expression is 0.374 = 1/0.728 - 1.

Various other things can also be calculated, knowing the Hubble time Y(t) in the past, for example the matter density (dark, ordinary, plus a minor contribution from radiation)

$$\rho = \frac {3}{8\pi G}{\left(\frac{1}{Y^2} -\frac {1}{Y_\infty ^2 }\right)}$$

As a check, the calculated matter density at some time in the past should agree with the redshift of light from a given moment in the past.

$$(1+z)^3 = \frac {\rho (then)}{\rho (now)}$$

 

[marcus]

Using George's suggestion of tanh solution for Y(t) involves adapting the timescale. Maybe I should have used forwards time all along, rather than lookback.
To use the tanh solution we need t, measured from the start of expansion (say in convenient d units), and we need the quantity 3t/2Y_∞ = 3t/326
Here 326 is simply twice our familiar 163 number for the longterm Hubble time.

Lookback time [Gy]        13.7     13.6     13.5  ...   2       1        0
Time since start [d]      0.573    1.57     2.57  ... 117.57  127.57   137.57
Y(t)=163tanh(3t/326) [d]  0.859    2.355    3.854 ... 129.41  134.56   138.99 
tanh estimate for z      44.6     22.3     15.8   ...   0.162   0.077    0.001

As a practical matter, to calculate z(t), using current estimates of the parameters, one could use
1 + z = [(coth(3t/326)² - 1)/0.374]^1/3
But the google calculator does not do hyperbolic cotangent, so instead of coth² one can use tanh^-2 and paste this in:
((tanh (.573*3/326)^-2 - 1)/.374)^.333
It gives 45.6 which is roughly right for 1+z, making z about 44.6

 

[Jorrie]

As a practical matter, to calculate z(t), using current estimates of the parameters, one could use
1 + z = [(coth(3t/326)2 - 1)/0.374]1/3
But the google calculator does not do hyperbolic cotangent, so instead of coth² one can use tanh^-2 and paste this in:
((tanh (.573*3/326)^-2 - 1)/.374)^.333
It gives 45.6 which is roughly right for 1+z, making z about 44.6

This is a nice relationship, for although the cosmo-calculator gives time t for a given redshift z, one often needs it in the other direction, which one must find by manual iteration.

BTW, do you know of a way of finding the event horizon for any cosmic time t, other than the integration to t_{infinity} that e.g. Tamara Davis used in eq. A.20 of her "Fundamental Aspects" paper:

\chi_c(t) = c \int_{t}^{t_{end}} \frac{dt&#039;}{R(t&#039;)}

This is comoving distance and her 'R' is the expansion factor 'a'. I suppose one can stop after some time, since the horizon distance will become practically a constant, but how far into the future is required?

 

[marcus]

This is a nice relationship, for although the cosmo-calculator gives time t for a given redshift z, one often needs it in the other direction, which one must find by manual iteration.

BTW, do you know of a way of finding the event horizon for any cosmic time t, other than the integration to t_{infinity} that e.g. Tamara Davis used in eq. A.20 of her "Fundamental Aspects" paper:

\chi_c(t) = c \int_{t}^{t_{end}} \frac{dt&#039;}{R(t&#039;)}

This is comoving distance and her 'R' is the expansion factor 'a'. I suppose one can stop after some time, since the horizon distance will become practically a constant, but how far into the future is required?

Tamara's thesis! I remember delving into that (as curious amateur) when it came out! What a kick! She helps one concretely visualize stuff far in future, like galaxes on their way out thru the CEH. I think she must be Lineweavers pride and joy. No! I don't know of a better way to find the CEH. I always just use the eventual limiting value of 16 billion ly. Proper.
George Jones might be willing to go into this with you. Or heck! how about writing Tamara Davis? Give her a link to your calculator (which she will recognize as a community service) and ask her a concise question. She has an automatic interest in helping you because it's good for the community to have accurate cosmo calculators online.
I think she is at Copenhagen now. Don't know her email but if you want to write and have any trouble finding it let me know.

I used your calculator to estimate the expansion age if one uses the two numbers 139 and 163 as basic input, as if they were exact. And your figure of 0.0000812 for omega_rad.
It came to 13.7574 billion
So then I calculated the redshift for the PRESENT, which should come out exactly zero
((tanh (137.574*3/326)^-2 - 1)/(1/.7272-1))^(1/3)

The reason for the .7272 is that the ratio 139/163 is almost precisely sqrt.7272.

I like the idea of a cosmic calculator where your inputs are the two relevant Hubble times,
Y_now and Y_∞
In a clumsy way, I can implement that with your calculator.
The condition of flatness is:
.7272
.2727188
.0000812
and 139 corresponds to a conventional Hubble rate number of 70.3463274

that is what google gives if you say "1/(13.9 billion years) in km/s per Mpc"
It will convert 1/(13.9 billion years) into conventional expansion rate units.

So for fun I put those three omega numbers, and that conventional Hubble rate number into your calculator and it said the expansion age is 137.574

So I calculated the redshift for that expansion age:
((tanh (137.574*3/326)^-2 - 1)/(1/.7272-1))^(1/3)
and it came out pretty close to zero, which is nice.
1+z came out 1.00013003035, according to google.

You could say it did credit to your calculator---that there is, somehow, not a lot of unnecessary roundoff error, and that the number 0.0000812 for radiation is OK. Neat.

I mean neat because, if you give your calculator inputs corresponding "exactly" to those two Hubble times (139 and 163) then it gives an age 137.574 which, if you check, MATCHES very well those two exact inputs. Using George's tanh and the google calculator to check the match. I think you understood what I was saying but I'm repeating in case other readers might not have.

It's really very nice. It means that if I want to use my juniorhigh school Hubble times as cosmic model inputs I simply have to prime your calculator with the numbers
.7272
.2727188
.0000812
70.3463274
and then use an expansion age of 137.574

 

[marcus]

Here's a cosmic history tabulation based on treating now and future Hubble times 13.9 and 16.3 billion years as exact. In the following calculations, 0.7272 is the square of their ratio, and 326 is twice 163.
Google codes used are:
z_t = ((tanh (t*3/326)^-2 - 1)/(1/.7272-1))^(1/3) - 1
Y_t = 163 tanh(t*3/326)
To calculate with a code, paste into google, replace t by an expansion age, and press = or return.

EXPANSION HISTORY, 139/163 MODEL.
Hubble time Y=1/H and age are shown in units of d = 10⁸ years.
Age at present 137.574 d.

   Age          Redshift         Hubble time 
   t (d)           z[SUB]t[/SUB]             Y[SUB]t[/SUB] (d)    
   0.0049         1093           0.00735
   1            30.574            1.5000 
   2            18.890            2.9997
   3            14.178            4.4989
   4            11.528            5.9973      
   5             9.796            7.4947
   6             8.559            8.9909
   7             7.625           10.4855
   8             6.889           11.9784
   9             6.292           13.4692
  10             5.796           14.9578
  20             3.269           29.6658
  30             2.243           43.8906
  40             1.659           57.4293
  50             1.273           70.1200
  60             0.992           81.8468
  70             0.776           92.5405
  80             0.603          102.1752
  90             0.459          110.7617
 100             0.337          118.3402
 110             0.232          124.9720
 120             0.139          130.7319
 130             0.057          135.7021
 131             0.049          136.1590
 132             0.041          136.6088
 133             0.034          137.0518
 134             0.026          137.4880
 135             0.019          137.9174
 136             0.012          138.3402
 137             0.004          138.7565
 137.574         0.000          138.9925

z : fractional amount distances and wavelengths have increased while light was in transit. Arriving wavelength is 1+z times original.
H=1/Y: Hubble expansion rate. Distances between stationary observers grow at this fractional rate--a certain fraction or percentage of their length per unit time.
H(per d) : fractional increase per convenient unit of time d = 10⁸ years.
Y=1/H: Hubble time. 1% of the current Hubble time is how long it would take for distances to increase by 1%, growing at current rate. At present, Y is 139 d = 13.9 billion years.
Hubble time is proportional to the Hubble radius = c/H: distances smaller than this grow slower than the speed of light. At present, the Hubble radius is 13.9 billion ly (proper distance)

The Hubble law describes the expansion of distances between observers at rest with respect to the background of ancient light and the expansion process itself: Observers who see the ancient light and the expansion process approximately the same in all directions, e.g. no Doppler hotspots.
The field of an observer's view can be thought of as pear-shape because distances were shorter back then. Here is a picture of an Anjou pear.
http://carrotsareorange.com/wp-content/uploads/2010/05/pear-anjou.jpg
Here is Lineweaver's spacetime diagram:
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure1.jpg
The upperstory figure, with horizontal scale in proper distance, shows the lightpear outline.
Here is Lineweavers plot of the growth of the scalefactor R(t), which models the growth of all distances between observers at universe-rest (at rest with respect to background.)
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure14.jpg
The dark solid line is according to standard model parameters. Various other cases are shown as well.