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

[Jorrie]

Here's a cosmic history tabulation based on treating the 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.

It is interesting to note that for time t > 1d, the errors for the assumption is less than 2%, while at t = 0.0049d, it grows to 10%. The 'real' redshift for t = .0049d is z = 950. I think the error is due to the assumption to ignore radiation density in the 139/163 model. It plays a significant role for z > 1000. At z ~3300, past radiation and matter energy densities were the same.

As long as we keep the redshift low enough, the 139/163 model is an excellent approximation and maybe worth a little calculator on its own...

 

[marcus]

YIPPEE! I would like to see such a thing!

I think you are right about the cause of the trouble at very early times and high redshifts.
The Y' prime equation (for which the tanh formula is a solution) was derived assuming matter-dominated era. The different behavior of radiation, during expansion, was not taken into account.

 

[marcus]

I guess what I would find most interesting would be a calculator where you do not put in a value for H_now or Omega_Lambda
but rather you give it two Hubble times to work with:
Y_now and Y_∞

But to improve performance at early times, what would be great is if you could put
in a third parameter as well, something like the present ratio of radiation to (dark and ordinary) matter.
And have the calculator take account of the balance shifting towards radiation in early times.

Basically I don't care so much about accuracy for z > 1000, but it would be nice to stay reasonably accurate back as far as z = 1000.
I noticed that for the time the CMB was emitted ( last scattering time), it was giving t = 490,000 years instead of t = 390,000 years.

I don't know how easy or hard it would be to implement.

I see your present default ratio of radiation to matter is 0.0000812/0.272 = 0.0002985

What I imagine being able to do is to decide on three parameters:
say 13.9 billion years
16.3 billion years
and present rad/mat ratio 0.0002985

So I enter those three parameters. And then being able to calculate a time from a redshift,
or (if it was the other way around) a redshift from a time.
Or maybe two outputs: an age and a Y, from a redshift.
or a redshift and a Y, from an age.

I wouldn't care about having the calculator output values of the Hubble rate, because I can always flip the Hubble time over and interpret 13.9 billion years as "1/139 percent per million years."
That's about as far as I've gone, thinking about it.

Anyway, it's an exciting idea. Any cosmo calculator that you prime basically by just putting in two Hubble times.

 

[marcus]

My viewpoint is more or less as follows: All distances between pairs of stationary observers have an intrinsic tendency to grow at a certain percentage rate, which we think is about 1/163 % per million years. Why this should be is an interesting question probably having to do with the microscopic structure of geometry, geometry at quantum level. This is simply observed, and speculated about, but not yet understood.

Geometry has its own dynamic rules and, once some event starts it expanding, tends to continue, subject to gradual slowing by matter. The present rate is larger than the intrinsic "vacuum" rate of expansion. We believe it is currently 1/139 % per million years. The excess is gradually being reduced, and to our best knowledge in the distant future, say 20 billion years hence, expansion will nearly be stabilized at 1/163 %.

The question on my mind is how to describe this to general audience, how to introduce expansion cosmology in a way that does not give people misleading pictures.

Like galaxies hurtling thru space at fantastic speeds, or like a mysterious "energy" pushing on them so that they accelerate. Such pictures, once gotten into the head, interfere with understanding and are difficult to root out.

I am exploring ways to introduce cosmology without ever showing the reader the Hubble constant in conventional km/s per Mpc terms. Because, when someone tries to get an intuitive grasp of H in those terms they are apt to fall into the trap of imagining it as the speed (km/s) of something moving thru space. It's the original pedagogical blunder.

People could learn to think about expansion of various distances as similar to the percentage growth of various savings accounts at bank. Different size accounts grow by different annual amounts--all growing at a certain interest rate.

Once you learn the Hubble growth rate H as 1/139 % per million years then LATER you might work with it as the speed (e.g. 70.35 km/s) with which a Megaparsec distance is growing. Nothing is moving thru space, but the distance between two stationary observers who are one Mpc apart is increasing at that speed. (One of the things dynamic geometry can do, just as it can make corners of a triangle add up to more or less than 180 degrees, depending.)

So suppose we want to put the cosmo-intro focus on the dynamics of a distance growth rate. How do we do this?

Well one nice way is to focus on the evolution of Y=1/H the Hubble time. I'm using uppercase Y rather than a T with subscript to keep notation simple. Uppercase Y looks like T with arms bent up.

Y grows according to a simple differential equation Y' = (3/2) (1- Y²/Y_∞²) involving dimensionless quantities, which applies back nearly to the start, assuming spatial flatness. It isn't valid back in the initial radiation dominated era but that is comparatively brief, so it works over much of expansion history. And the equation has an explicit solution Y(t) = 163 tanh(3t/326).

The Hubble time constant 16.3 billion years carries the same information as the usual cosmological constant, and in the above formula the 163 appears twice. Once as coefficient out in front of the tanh and again, doubled, as 326 in the argument. The pedagogical idea here is that beginners get hands-on experience with this cosmo constant information as a feature of geometry that they can calculate with---a definite and inherent geometric feature of spacetime that they can use with a calculator, and not as some mysterious "dark energy" constituting a possibly fictitious fraction of the "critical density".

Thanks to G. Jones, Jorrie, and Mark M. for help and encouragement with this.

 

[Jorrie]

So suppose we want to put the cosmo-intro focus on the dynamics of a distance growth rate. How do we do this?

Well one nice way is to focus on the evolution of Y=1/H the Hubble time. I'm using uppercase Y rather than a T with subscript to keep notation simple. Uppercase Y looks like T with arms bent up.

Y grows according to a simple differential equation Y' = (3/2) (1- Y²/Y_∞²) involving dimensionless quantities, which applies back nearly to the start, assuming spatial flatness. It isn't valid back in the initial radiation dominated era but that is comparatively brief, so it works over much of expansion history. And the equation has an explicit solution Y(t) = 163 tanh(3t/326).

Well motivated, Marcus.

One thing that still bothers me a little is the units d=10^8 years that you are using. It's still somewhat confusing and I cannot quite remember your motivation for not using standard 10^6 or 10^9 years. Things should work equally well with 13.9 and 16.3 Gy, or perhaps 13900 and 16900 My.

I think it should be simple to include radiation density, but I will experiment a bit. I suspect that the growth rate law during the radiation dominated phase changes to something like
Y' = 2 (1- Y_{rad}^2/Y^2), but I need to verify this.

 

[marcus]

Matter era: Y' = \frac{3}{2}(1- Y^2/Y_\infty^2)
Radiation era: Y' = 2 (1- Y^2/Y_\infty^2)
I will double check that in the next couple of posts.
Jorrie you are right about the odd time unit. It is an ignominious practice to introduce a working time unit. I'm thinking "d" stands for "deci" as in deciBell, deciLiter, deciMeter. I just find it very comfortable to work in that size (a tenth of a billion years) time unit while we are developing this approach but in the end it will probably be changed to giga year (Gy) or mega year.

Now we've reached the point where we need to consider radiation density. So I will quote the overly verbose post #360 which has some stuff about deriving the Y' equation and we will see how it is different in the radiation-era, compared with the matter-era. I think your coefficient of 2 in the preceding post, instead of 3/2 is right. In fact you must be somewhat of an expert in this kind of thing, I expect, having built your calculator. But I want to go thru it myself. First I will quote this incredibly long post
Scroll down to the RED highlight.
===quote post #360===
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...

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====
...
==endquote==

 

[marcus]

So we have to change this:

Differentiating, we get an important formula for the change in density, namely ρ'.
ρ' = (M/a³)' = -3(M/a⁴)a' = -3ρ(a'/a) = -3ρH
​

into this:

Differentiating, we get an important formula for the change in density, namely ρ'.
ρ' = (M/a⁴)' = -4(M/a⁵)a' = -4ρ(a'/a) = -4ρH
​

And we have to change this:

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
​

into this:

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)(-4ρH) = - (32/3)πGρH, and we can cancel 2H to get the change in H, namely H':

H' = - (16/3)πGρ
​

And finally, we have to change this:

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]
​

into this:

Y' = (1/H)' = - H'/H² = (16/3)πGρ/H² = (16/3)πGρY²

All this uses is H' = - (16/3)πGρ, 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' = (16/3)πG (3/8πG)[1/Y² - 1/Y_∞²]Y²


Y' = 2[1/Y² - 1/Y_∞²]Y²
= 2[1- Y²/Y_∞²]​

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

Yes, you were right about the coefficient. The arithmetic is simply that 16/3 x 3/8 = 2.

Hot dog! (sorry about long-windedness, old guys have to proceed deliberately, so easy to make mistakes )

The real hard thing, I think, is when the radiation and matter densities are around the same order of magnitude, neither one dominant. Then it seems like a BLEND of the two different Y' equations.

 

[marcus]

I guess one way to do it would be simply to DECLARE that the radiation era lasted until, say, 300,000 years, and up to that point go with Y(t )= 163 tanh(t*2/163)

and then at that point in time, which I'm still calling 0.003 d, you switch over to using
Y(t )= 163 tanh((t+.001)*1.5/163)
where the time advance of 0.001 d is for continuity. So that one function takes over where the other left off. They match, at transition time, with that adjustment.
Y(t )= 163 tanh((.003+.001)*1.5/163) = 163 tanh((.003)*2/163)
So then let's see how things look at the last scattering time of 390,000 years (0.0039 d):

Y(.0039 )= 163 tanh((.0039+.001)*1.5/163) = 0.0735

Not bad! It's sort of an OK Hubbletime for last scatter.
===================
And here's the corresponding redshift:
((tanh ((.0039+.001)*1.5/163)^-2 - 1)/(1/.7272-1))^(1/3)
gives 1094
===================
It's not really satisfactory. The equation is really too simple to deal properly with early universe. Conventionally I think one says radiation era lasts briefer, e.g. to 54,000 years. Presumably there is a gradual transition with the large amount of energy in the form of light playing a significant role.
So I see no clearcut place where you change over from coefficient 2 to 1.5.
Just declaring a transition at time 300,000 is a kludge. But with such simple tools, and limited possibilities, it might be the best way out.

 

[Jorrie]

I guess one way to do it would be simply to DECLARE that the radiation era lasted until, say, 300,000 years, and up to that point go with Y(t )= 163 tanh(t*2/163)

and then at that point in time, which I'm still calling 0.003 d, you switch over to using
Y(t )= 163 tanh((t+.001)*1.5/163)
...
It's not really satisfactory. The equation is really too simple to deal properly with early universe. Conventionally I think one says radiation era lasts briefer, e.g. to 54,000 years. Presumably there is a gradual transition with the large amount of energy in the form of light playing a significant role.
So I see no clearcut place where you change over from coefficient 2 to 1.5.
Just declaring a transition at time 300,000 is a kludge. But with such simple tools, and limited possibilities, it might be the best way out.

I also experimented a bit on existing spreadsheets as a reference, but with similar mixed success. Could get it close to right for the CMB-era and for Now, but with uncomfortable deviations en-route.

I woke up (yes, it's rise and shine time here already), with the subconscious telling me the following: convert the new inputs to the old inputs behind the scenes and perform the proper Friedman calculation à la the old calculator. Then pump out a simplified set of results and give your approximation equations in info popups, with some caveats.

How would this sit with you?

PS: It's also a lot less work... ;-)

 

[marcus]

I imagine I would be delighted.
The main requirement is that the project make sense to you (the calculator builder) and that you be satisfied with the results.

I was intrigued by the idea of a calculator (possibly quite simple) that would not outwardly involve H and Omega_Lambda. It would let you control the current growth rate and the future growth rate by entering two Hubbletimes, instead.

Anything that does this, however it does it, seems like an interesting pedagogical tool. No outward reference to "km/s" and "dark energy". Instead: a current percentage growth rate and a future asymptotic one.
I think Hubbletimes are probably the easiest handles to use, to specify current and future growth rates.
So I immediately think of being able to input, say, 13.9 Gy and 16.3 Gy. and maybe the presentday ratio of matter to radiation, and that's it. After that I can convert any redshift to several outputs, or an expansion age.

But you are the one who has worked on cosmology calculators so you will have your own criteria and ways to reckon how well things will communicate to the user. You're the one with experience, so you be the judge.

As I recall I found that 13.9 and 16.3 corresponded to something like 70.35 km/s per Mpc and
.7272 for Omega_Lambda.
Then Omega matter was 0.2727188
And Omega radiation was 0.0000812
So they added up to 0.2728, giving flatness.

That means the matter/radiation ratio was 3359*. (Which is why you need to go back to a redshift of around 3350 or 3360 in order for them to be on par with each other.)

So if I was using your calculator I would like to be able to input 13.9 billion and 16.3 billion and a number like 3360.
Then the calculator would secretly change (13.9, 16.3, 3360) inputs for the existing program and proceed from there.
I'm getting curious to see how this takes shape! It is like constructing a new "front end" for something you already have that runs well.

Let me check that my memory was accurate about that 70.35...
I paste this into google:
1/(13.9 billion years) in km/s per Mpc
Yes! it immediately comes back with "70.3463274 (km/s) per Mpc"
Since it is internal, perhaps better to use 70.34633 or something like that.
And google also tells me that 13.9^2/16.3^2 = 0.727200873

*calculated from 2727188/812=3358.6...

 

[marcus]

Updated expansion table based on treating now and future Hubbletimes 13.9 and 16.3 billion years as exact. In the following calculations, 0.7272 is the square of their ratio, and a time unit (d) is used which is a tenth of a billion years. Age at present 137.574 d. The model has to be considered a "toy" because the simplified equations give only rough approximation at times before 1 d. I'm trying out an additional column for the scalefactor a_t = 1/(1+z_t) which shows the growth of a generic distance.
Google codes used are:
a_t = (1/.7272-1))^(1/3)/((tanh((t+.001)*1.5/163))^-2 - 1)^(1/3)
z_t = ((tanh((t+.001)*1.5/163))^-2 - 1)/(1/.7272-1))^(1/3) - 1
Y_t = 163 tanh((t+.001)*1.5/163)
To calculate with a code, paste blue expression into google, replace t by an expansion age, and press =.

EXPANSION HISTORY, 139/163 MODEL.

   Age          Redshift        Hubble time        Scale factor
   t (d)           z[SUB]t[/SUB]             Y[SUB]t[/SUB] (d)               a[SUB]t[/SUB] 
   0.0030         1252           0.00600             0.0008  
   0.0039         1093           0.00735             0.0009
   1            30.553            1.501              0.032
   2            18.883            3.001              0.050
   3            14.175            4.500              0.066
   4            11.526            5.999              0.080
   5             9.794            7.496              0.093         
   6             8.558            8.992              0.105
   7             7.624           10.487              0.116
   8             6.888           11.980              0.127
   9             6.291           13.471              0.137
  10             5.796           14.959              0.147
  20             3.269           29.667              0.234
  30             2.243           43.892              0.308
  40             1.659           57.431              0.376
  50             1.273           70.121              0.440
  60             0.992           81.848              0.502
  70             0.776           92.542              0.563
  80             0.603          102.176              0.624
  90             0.459          110.762              0.685
 100             0.337          118.341              0.748
 110             0.232          124.973              0.812
 120             0.139          130.732              0.878
 130             0.057          135.703              0.946
 131             0.049          136.159              0.953
 132             0.041          136.609              0.960
 133             0.034          137.052              0.967
 134             0.026          137.488              0.974
 135             0.019          137.918              0.981
 136             0.012          138.341              0.989
 137             0.004          138.757              0.996
 137.574         0.000          138.993              1.000

For times earlier than 0.0030 d (before year 300,000) these "radiation era" google codes are preferable:
z_t = ((tanh(t*2/163)^-2 - 1)/(1/.7272-1))^(1/3) - 1
Y_t = 163 tanh(t*2/163)
The same caveat applies. Only a rough approximation to early universe behavior.
Some notes on the table:
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.

 

[Jorrie]

So if I was using your calculator I would like to be able to input 13.9 billion and 16.3 billion and a number like 3360.
Then the calculator would secretly change (13.9, 16.3, 3360) inputs for the existing program and proceed from there.
I'm getting curious to see how this takes shape! It is like constructing a new "front end" for something you already have that runs well.

Yes, I have similar ideas, but I now tend towards using three Hubble times: start of the matter era (Y_m=0.1 My)*, the present day (Y_0=13900 My) and the maximal Hubble time (Y_inf=16300 My) as default constants (changeable by user). Then let the user specify a range of either z or t, with a required incremental step. The calculator then to produce one of your tables automatically. The project requires redesigning and programming new input and output ends for the existing 'LCDM engine'. Will see how far this takes us...

*Start of the matter era is at radiation/matter equalization around z=3300, with t=0.057 My and Hubble time 0.1 My. This is enough info to bring in the radiation energy effect accurately.

 

[marcus]

Yes, I have similar ideas, but I now tend towards using three Hubble times: start of the matter era (Y_m=0.1 My)*, the present day (Y_0=13900 My) and the maximal Hubble time (Y_inf=16300 My) as default constants (changeable by user). Then let the user specify a range of either z or t, with a required incremental step. The calculator then to produce one of your tables automatically. The project requires redesigning and programming new input and output ends for the existing 'LCDM engine'. Will see how far this takes us...

*Start of the matter era is at radiation/matter equalization around z=3300, with t=0.057 My and Hubble time 0.1 My. This is enough info to bring in the radiation energy effect accurately.

An extremely interesting idea. IMHO you could design a whole semester course around that kind of teaching/learning resource.

It is intriguing to think of a calculator that you put 3 model parameters into and it then generates a table, going along the t-scale step by step.

I would experiment with using the scalefactor as an alternative lefthandcolumn variable, instead of z.

The thing is, when someone says we see this galaxy with redshift 4, if you want to look it up you could just think: scalefactor a = 1/(1+z) = 1/5 = 0.2
I am looking at this galaxy as it was when distances were 20% of what they are today.
that galaxy I'm looking at is back in the days of scalefactor 0.2

I would want undergrad students to be familiar with converting z that they read into scalefactor, and then putting scalefactor into calculator.

So I would put the scalefactor along the lefthand column of the second option table. And next to it the time (derived from that scalefactor).

IMO we observe the scalefactor just as directly (from the spectrum of incoming light) as we observe the z. they are just different algebraic versions of the same basic datum.

And a is increasing, it is a lot more like t. You've got to follow your own craft-sense. But I think I'll try making a table with increments of scalefactor a and see what it looks like.
Maybe it's a bad idea for some reason I don't see yet.

Your idea of making something that will accept 3 inputs like (13900, 16300, 0.1 My) and from those 3 inputs crank out a table (even a small table, with specified range and stepsize) is terrific.
==================
EDIT: have to go to the trainstation but just want to write down this google code (no time to check it)
t+0.001 = (163/1.5)arctanh sqrt(a^3/(a^3 -1 + 1/.7272))

EDIT: google calculator does not have arctanh, or artanh, the inverse of tanh. I will try to implement using the analytical expression for arctanh, which employs the natural logarithm ln(x)
t+0.001 = (163/3)ln(1+(1+(1/.7272-1)/a^3)^-.5) - (163/3)ln(1-(1+(1/.7272-1)/a^3)^-.5)

When I try this with a = .5 I get that t+.001 = about 59.7. Seems right, so I'll make a table based on the scalefactor.

Scalefactor   Age Gy
.1                0.56
.2                1.58
.3                2.88
.4                4.37 
.5                5.97
.6                7.61
.7                9.24 
.8               10.82  
.9               12.33
1.0              13.759             
1.1              15.11
1.2              16.38
1.3              17.57
1.4              18.70

Scalefactor 1.4 refers to a time in the future when they will observe OUR light with wavelengths 140% of what they were when our stars emitted the light, today. Somewhere in some galaxy they will point a telescope at the Milkyway and see light emitted by the sun and other stars today. And the wavelength will be extended by a factor of 1.4.
The table shows that that will happen about 5 billion years from now.

 

[Jorrie]

I would experiment with using the scalefactor as an alternative lefthandcolumn variable, instead of z.
...
EDIT: google calculator does not have arctanh, or artanh, the inverse of tanh. I will try to implement using the analytical expression for arctanh, which employs the natural logarithm ln(x)
t+0.001 = (163/3)ln(1+(1+(1/.7272-1)/a^3)^-.5) - (163/3)ln(1-(1+(1/.7272-1)/a^3)^-.5)

When I try this with a = .5 I get that t+.001 = about 59.7. Seems right, so I'll make a table based on the scalefactor.

Scalefactor   Age Gy
.1                0.56
.2                1.58
.3                2.88
.4                4.37 
.5                5.97
.6                7.61
.7                9.24 
.8               10.82  
.9               12.33
1.0              13.759             
1.1              15.11
1.2              16.38
1.3              17.57
1.4              18.70

As a matter of fact, most cosmo-calculators use a as the core independent variable that they ramp up or down, normally for a from ~0 to 1, i.e. the best part of post-inflation expansion history. My calculators do the same. It is simply easier to set it up so that a=1 (identically) comes out of the numerical integration.

The equation that you use will again be very close for a > .01, but will start to deviate for smaller a, due to the hotter radiation at early times. Our proposed simplified calculator should be fairly accurate for a down to around one millionth or so, provided that we get the radiation component in correctly.

 

[marcus]

As a matter of fact, most cosmo-calculators use a as the core independent variable that they ramp up or down, normally for a from ~0 to 1, i.e. the best part of post-inflation expansion history. My calculators do the same...

...down to around one millionth or so, provided that we get the radiation component in correctly.

That's good news. As I see it a is a directly observed quantity. z is just an algebraic variant of a. When you look at the hydrogen line in a spectrograph and see by what ratio the wavelength is enlarged you could just as well consider that you are reading a off the instrument as think of it as reading z, which is just z = 1/a - 1.

So a is a directly observed (not model dependent) quantity, and it is also a key variable in the calculation. The fact that it's this way gives IMO a solid empirical feel to the situation.

I think in my dream calculator you would have a box for 1+z, and a box for a. They are reciprocals of each other, and putting a number in either would work. It wouldn't have a box for z. If a student reads somewhere that a galaxy was observed with redshift 3, then he or she should know to put in 4, or mentally convert that to 1/(3+1) = 0.25 and put 0.25 into the a box.

It's getting late here. Maybe some fresh ideas in the morning. I should try to make this more compact:

t+0.001 = (163/3)ln(1+(1+(1/.7272-1)/a^3)^-.5) - (163/3)ln(1-(1+(1/.7272-1)/a^3)^-.5)

t_a = (16.3/3)ln((1+(1+(1/.7272-1)/a^3)^-.5)/(1-(1+(1/.7272-1)/a^3)^-.5))

I've dropped the little time adjustment of .1 million year, and put a decimal point into the 16.3 so it gives answers in billions of years as in that brief table.
So now, associated with every directly measurable scalefactor a we have the estimated expansion age TIME when distances were that size, or when the light was emitted.
And our handle on how fast the world was expanding at that epoch is the Hubbletime. Basically a sort of linear "doubling time" for distance growth. To every scalefactor a in the past there should be an associated growthrate.
Let's add Hubbletime Y_a to that brief table:
Y_a= 16.3(1+(1/.7272-1)/a^3)^-.5

Scalefactor     Age (Gy)          Hubbletime (Gy)        ?
 a                 t[SUB]a[/SUB]                  Y[SUB]a[/SUB]                Δ[SUB]a[/SUB]
.1                0.56                 0.84              5.38  
.2                1.58                 2.36              3.46
.3                2.88                 4.22              2.61         
.4                4.37                 6.22              2.11
.5                5.97                 8.15              1.77
.6                7.61                 9.85              1.53
.7                9.24                11.26              1.35
.8               10.82                12.38              1.21
.9               12.33                13.24              1.09
1.0              13.759               13.900             1.00        
1.1              15.11                14.40  
1.2              16.38                14.77
1.3              17.57                15.06
1.4              18.70                15.29

So, to read something off the table, it says that a little over 2 billion years from now there will be people in another galaxy looking at our Milkyway galaxy with their telescope and they will observe that the hydrogen wavelengths are 30% longer (than hot hydrogen rainbow wavelengths in their lab) and they will say "Hmmm, distances back then when the light was emitted were 1/1.3 what they are today..." And they will be wondering how long ago that was rapidly distances were expanding back then so they will look at their table and say "Hmmm, that was 2.2 billion years ago, and in those days it took only 139 million years for a distance to grow 1%, whereas now it takes 150.06 million years, so expansion was more rapid back then."

While I can still edit I'll try adding an interesting incremental distance number that can be calculated at each scalefactor a:

Δ_a = 1/sqrt(.2728*a + .7272*a^2)
It may turn out to have no use, but the table has room for another column

 

[Jorrie]

*Start of the matter era is at radiation/matter equalization around z=3300, with t=0.057 My and Hubble time 0.1 My. This is enough info to bring in the radiation energy effect accurately.

I had a look at this from the Friedman POV. All we need are the three Hubble times: rad/matter equality, Y_eq = 0.1 My, Y_now = 13900 My and Y_inf = 16300 My, plus the redshift for equality, z_eq = 3350. Assuming flatness, all three present energy densities are then calculable. The rest is just a matter of standard calculation and presentation.

The most troublesome one to find from the inputs is the present radiation energy density, but because it is very small (relatively), the following seems to work well:
\Omega_r = \left(\frac{Y_{now}}{Y_{eq}}\right)^2 a^4 - \Omega_m a - \Omega_\Lambda a^4
where: a=(1+z_{eq})^{-1}, \Omega_\Lambda = (Y_{now}/Y_{inf})^2, \Omega_m\approx 1-\Omega_\Lambda (provided \Omega_r \ll 1).

I have checked this by means of a spreadsheet and it looks promising, with errors far below the input accuracies throughout the redshift range of interest, zero to 3350.

Edit: Surprisingly, the rather complex equation may be unnecessary, because a simple \Omega_r \approx \Omega_m/z_{eq} seems to be just as accurate.
Marcus mentioned this relationship in a prior reply.

 

[marcus]

Jorrie, I didn't see your last post (#416) until just now when I posted mine. I'm glad to hear this, it's looking good!

I had a look at this from the Friedman POV. All we need are the three Hubble times: rad/matter equality, Y_eq = 0.1 My, Y_now = 13900 My and Y_inf = 16300 My, plus the redshift for equality, z_eq = 3350. Assuming flatness, all three present energy densities are then calculable. The rest is just a matter of standard calculation and presentation.

The most troublesome one to find from the inputs is the present radiation energy density, but because it is very small (relatively), the following seems to work well:
\Omega_r = \left(\frac{Y_{now}}{Y_{eq}}\right)^2 a^4 - \Omega_m a - \Omega_\Lambda a^4
where: a=(1+z)^{-1}, \Omega_\Lambda = (Y_{now}/Y_{inf})^2, \Omega_m\approx 1-\Omega_\Lambda (provided \Omega_r \ll 1).

I have checked this by means of a spreadsheet and it looks promising, with errors far below the input accuracies throughout the redshift range of interest, zero to 3350.

I added to that brief table based on the scalefactor.
It's intended to be the 13.9/16.3 model we've been concentrating on and I think the first three columns are right, but am not sure about the accuracy of the last two, the distances to a source at the given scalefactor.
EDIT: For clarity I will write out the google calculator expression for t_a in LaTex:
t_a = \frac{16.3}{3}ln\left((1+(1+(1/.7272-1)/a^3)^{-.5})/(1-(1+(1/.7272-1)/a^3)^{-.5}) \right)
Here's the expression as used in the calculator:
t_a = (16.3/3)ln((1+(1+(1/.7272-1)/a^3)^-.5)/(1-(1+(1/.7272-1)/a^3)^-.5))
Here's the expression for the Hubble time:
Y_a= 16.3(1+(1/.7272-1)/a^3)^-.5

Scalefactor  Age (Gy)    Hubbletime (Gy)   Proper distance to source (Gly)
 a    1/a-1    t[SUB]a[/SUB]            Y[SUB]a[/SUB]               D[SUB]now[/SUB]         D[SUB]then[/SUB]
.1    9.0     0.56          0.84              30.9         3.09
.2    4.0     1.58          2.36              24.0         4.79
.3    2.333   2.88          4.22              18.7         5.62       
.4    1.5     4.37          6.22              14.5         5.79
.5    1.0     5.97          8.15              10.9         5.45
.6    0.666   7.61          9.85               7.9         4.74
.7    0.428*  9.24         11.26               5.4         3.78
.8    0.25   10.82         12.38               3.3         2.63
.9    0.111  12.33         13.24               1.5         1.36
1.0   0.0    13.759        13.900              0.00        0.00  
1.1          15.11         14.40  
1.2          16.38         14.77
1.3          17.57         15.06
1.4          18.70         15.29

*0.428571429
(13.9 Gy, 16.3 Gy, flat) → (70.3463, 0.7272, 0.2728)

 

[Jorrie]

It's intended to be the 13.9/16.3 model we've been concentrating on and I think the first three columns are right, but am not sure about the accuracy of the last two, the distances to a source at the given scalefactor.

The "13.9/16.3 model" is perfectly accurate for the scalefactors that you have shown. It's only from a < 0.01 that accuracy becomes an issue due to radiation density.

I also like the scalefactor "input column", because one can go as far into the future as desired ( a > 1). If we use redshift, it would be negative for the future and that's an awkward concept.

PS: look at the edit I've made to post #416.

 

[marcus]

I found out a minor detail about Wright's calculator. when you tell it .7272 and .2728 it actually uses those values, although it REPORTS that it is using ..727 and .273.

IOW it rounds off what it says the model parameters are that it is using, but you can see the difference in the results. It's just a minor thing, but it's convenient.

You can actually get that calculator to use (70.3463, .7272, .2728) even though it may look as if you can't (because of this rounding off.)

I saw the edit in #416, thanks for the mention :-)
it makes sense. That aspect (getting the right radiation component) looks very hopeful.
What I'm not sure about is how you will be able to build a different "front end"

EDIT: For clarity I will write out the google calculator expression for t_a in LaTex:
t_a = \frac{16.3}{3}ln \frac{1+(1+\frac{1/.7272-1}{a^3})^{-.5}}{1-(1+\frac{1/.7272-1}{a^3})^{-.5}}
t_a = \frac{16.3}{3}ln \frac{1+(1+(1/.7272-1)/a^3)^{-.5}}{1-(1+(1/.7272-1)/a^3)^{-.5}}
t_a = \frac{16.3}{3}ln\left(\frac{(1+(1+(1/.7272-1)/a^3)^{-.5})}{(1-(1+(1/.7272-1)/a^3)^{-.5})} \right)
t_a = \frac{16.3}{3}ln\left((1+(1+(1/.7272-1)/a^3)^{-.5})/(1-(1+(1/.7272-1)/a^3)^{-.5}) \right)

Finally, here's the expression I paste into google calculator for t_a:
(16.3/3)ln((1+(1+(1/.7272-1)/a^3)^-.5)/(1-(1+(1/.7272-1)/a^3)^-.5))

Here's the corresponding expression for the Hubble time Y_a:
16.3(1+(1/.7272-1)/a^3)^-.5

 

[Jorrie]

That aspect (getting the right radiation component) looks very hopeful.
What I'm not sure about is how you will be able to build a different "front end"

If one ignores the small curvature caused by the present radiation density when you determine the radiation density parameter for matter equality, the 'front-end' is actually straightforward. From Y_now, Y_inf and z_eq, the three energy densities are as before:
\Omega_\Lambda = (Y_{now}/Y_{inf})^2; \Omega_m \approx 1-\Omega_\Lambda; \Omega_r\approx \Omega_m /(z_{eq}+1) and H_0 = 1/Y_{now} of course.
This we send to the full version's numerical integration module. Strictly speaking, we should also input the cosmic time (t) for r-m equality, but provided we start the integration early enough (well before r-m equality), we can set the starting time to zero.

EDIT: For clarity I will write out the google calculator expression for t_a in LaTex:
t_a = \frac{16.3}{3}ln\left((1+(1+(1/.7272-1)/a^3)^{-.5})/(1-(1+(1/.7272-1)/a^3)^{-.5}) \right)

Finally, here's the expression I paste into google calculator for t_a:
(16.3/3)ln((1+(1+(1/.7272-1)/a^3)^-.5)/(1-(1+(1/.7272-1)/a^3)^-.5))

Here's the corresponding expression for the Hubble time Y_a:
16.3(1+(1/.7272-1)/a^3)^-.5

I've included this in the draft spreadsheet for the "lean model" that I mailed to you for comment. The spreadsheet shows your time approximations to be within 1.5% for z < 100, 15% for z < 1100 and 40% at r-m equality, good enough for early learning purposes.

 

[marcus]

My first reaction is (cheers!) we can throw out my google-calculator approximations for anything like z>> 10. They become too inaccurate for z > 100.

But I don't have microsoft Excel on this computer and have never installed a spreadsheet in my life (our son, who visits now and then, may help with that). So I don't have much of an idea how the new "front-end" will look and work.

I have the XLS file on my desktop, waiting, but so far have only opened it as text. I think I need Excel to open it as an actual spreadsheet.

This is exciting, I picture that the three inputs to the front end are (y_now, y_∞, z_eq) and that it outputs perhaps single values of stuff (like a, t, z, y_a or y_t, D_now, D_then...)
Or perhaps, if not now then possibly in future, a table. Assuming the user has specified a sequence of values of a, or values of t, to run down the first or lefthand column of the table.

That seems pedagogically beautiful, to me. It says to the beginner "all you do is specify two percentage growth growth rates of distance: the present and the eventual future one"
and the model does the rest.
So attention is focused on percentage growth rate instead of "speed".
And the cosmo constant is no mystery, but simply manifest in the eventual percentage growth rate.

Looking forward to seeing the actual front-end, this is just how I picture it.

 

[marcus]

In the meantime, I'm getting a routine down for setting up the 13.9/16.3 model in Jorrie's calculator.

First go to google and say Mpc/(km/s)/13.9 billion years

that outputs the number 70.3463274​

so you copy to clipboard and paste into the calculator's H_now box.

Next go to google and say 1 - .7272 - .0000812

that outputs the number 0.2727188​

so you copy that to clipboard and paste in the Omega Matter Now box.

Then you are ready to go! You have almost exactly the right parameters loaded for the 13.9/16.3 model. And if you want to try variations with different y_now and y_∞ you can use this same format. The only difference is the second step you say
1 - (y_now/y_∞)² - .0000812

Because all the .7272 is is the square of the ratio of the two Hubbletimes.

======================
Extra, probably unneeded explanation: the second step is to ensure perfect flatness.
Premultiplying 1/(13.9 billion years) by the factor Mpc/(km/s) is simply to get rid of the units km/s/Mpc so that what you get out is a pure number 70.346... to paste into the H box. In this approach
1/(13.9 billion years) actually is the Hubble growth rate, expressed in "per time" terms.

 

[Jorrie]

...
But I don't have microsoft Excel on this computer and have never installed a spreadsheet in my life (our son, who visits now and then, may help with that). So I don't have much of an idea how the new "front-end" will look and work.

It is possible that Firefox or MS IE may open the spreadsheet for you. I will also see if I can save the spreadsheet as a Google doc.

In the meantime I have attached the spreadsheet data as a .pdf, where you can see the 'front-end', but unfortunately not manipulate it. I could not get my pdf writer to print headers on each page; sorry about that, but it will give you a good idea.

The spreadsheet is primitive, but you will see some pointers like "equal", CMB for the green hi-lighted rows...

PS: I have also attached the graph for some values on the sheet.

 

[Jorrie]

And if you want to try variations with different y_now and y_∞ you can use this same format. The only difference is the second step you say
1 - (y_now/y_∞)² - .0000812

Because all the .7272 is is the square of the ratio of the two Hubbletimes.

I found the extra precision gained for 'perfect flatness' by the process you described to be absolutely negligible, even in the precise calculator. That's why the default Omega values in my calculator does not add up to precisely one, but to 1.0000812. This gives a very-very slight positive spatial curvature and hence a large, but finite cosmos (which I always like best :-)

 

[marcus]

More to my taste too. So be it.

The graph is handsome. Nice to see the Hubbleradius and the CEH coverging at 16.3 but having a temporary gap. Beautiful curves.
Also the red curve, if you flip the graph over, exchanging x and y axes, so time is on the horizontal and scalefactor is up the side, you get this Lineweaver figure #14
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure14.jpg

The nice thing is you can see the inflection in the curve, where it changes from convex to concave:
from decelerating to accelerating.

So in your figure you can see the same inflection, a bit below the 10 billion year line, we know it is about 7 billion, but it's nearly linear for a stretch so it's hard to spot the exact point of inflection by eye.

 

[marcus]

I should summarize where I see this going as a result of the work and discussion in this thread. We have two resources shaping up. On one hand there are Jorrie's calculator(s) which AFAICS deliver professional-grade accuracy and implement the standard cosmic model. One reason they are interesting is that they let you input different model parameters and they output quite a variety of information about the universe: distances to galaxies then and now, distance expansion speeds, percentage expansion rates, and so forth.

On the other hand we've got a simple "do it yourself" cosmic model that delivers reasonably good precision back over the past 12 or 13 billion years. It is more trouble to use because it is essentially just based on a single formula. This is the formula for the expansion history of a generic distance: the scalefactor a(t) as a function of time.

This is arbitrarily pegged at the present value of 1, so a(now) = 1, and it shows how any cosmological distance has grown---it also extends into the future to show expected future growth.

This simple DIY cosmic model is is more trouble to use because essentially given the growth history you have to figure everything else from that, for yourself. But it's not as bad as it sounds and one picks up some understanding along the way.

My posts on the last several pages can be summed up streamlined as follows.
If you go to the online calculator http://web2.0calc.com/ (the first hit if you do a search for "online scientific calculator")
and paste this formula in:

(((16.3/13.9)^2 - 1)/((tanh(1.5*t/16.3))^-2-1))^(1/3)

you will see that as soon as it is pasted in, the calculator displays a neater, easier-to-read form.\left(\frac{(\frac{16.3}{13.9})^2-1}{(tanh(\frac{1.5}{16.3}t))^{-2}-1}\right)^{1/3}What this computes, given a time t (in billion years) is the scalefactor a(t)---in other words it tells you the expansion history of a generic distance over time.
To use it, just replace the "x" by a time expressed in billions of years (such as 1, or 2, or 13.759 which is the current expansion age) and press the equals sign.
The answer will appear in the window and you can copy it to clipboard if you want. Then to repeat the calculation, click on the neat version of the formula as it appears above the window, and you can substitute something else in for the variable.
========================
1--- 0.1471433... (when the universe was 1 billion years old, distances were 14.7% what they are now)
2--- 0.2342347... (at 2 billion years, distances were about 23% what they are now)
...
...
13.759--- 0.9999836... (at the present age of 13.759 billion years they are of course 100% of their present lengths.:-)
...
20--- 1.5235746... (at age 20 billion years, distances will be 52% bigger than they are today.)
==================
A modified version of this same formula gives you the reciprocal scalefactor, 1/a(t), which turns out to be quite useful
The model's formula for 1/a(t) is
\left(\frac{(\frac{16.3}{13.9})^2-1}{(tanh(\frac{1.5}{16.3}t))^{-2}-1}\right)^{-1/3}
and the single-file version, that you paste into the calculator if you are working with it that way is:
(((16.3/13.9)^2 - 1)/((tanh(1.5*t/16.3))^-2-1))^(-1/3)

Adding up successive values of 1/a as you work back in time is actually the way the present distance to a source is calculated!
========================
Some examples of the reciprocal scalefactor 1/a(t) for various times.
1---6.80... (distances and wavelengths of traveling light have expanded by a factor of 6.8 since the universe was 1 billion years old
2--- 4.27... (distances and wavelengths of traveling light have expanded by a factor of 4.27 since the universe was 1 billion years old)
...
...
13.759--- 1.000... (this is the present, distances and waves are their present lengths :-)
...
==================
It's conceivable you might sometime want to find the time t that gives a particular scalefactor, IOW invert the above formula for a(t). In that case paste this in
(16.3/1.5)atanh((((16.3/13.9)^2 -1)/a^3+1)^-.5)
which the calculator will display as \frac{16.3}{1.5}atanh\left(\left(\frac{(\frac{16.3}{13.9})^2-1}{a^3}+1\right)^{-.5}\right)
==================
The main equation in this model is this one. It gives the scalefactor a(t) at each time, going back pretty far into the early history of expansion where it gets a bit off track (because when you get back to the first few 100 million years much of the density in the universe was radiation rather than particles of matter, and radiation behaves differently in expansion , so the physics is not as simple. Anyway the main model equation is this:

\left(\frac{(\frac{16.3}{13.9})^2-1}{(tanh(\frac{1.5}{16.3}t))^{-2}-1}\right)^{1/3}

The coefficient 1.5 and the exponent 1/3 both reflect the fact that we're in a matter dominated era, and have been since the first few 100 million years, and matter density falls off as volume increases---as the cube of distance. That is where the 3 and the 1.5 come from. In a radiation dominated world they would be 4 and 2. I derived some equations earlier in this thread and can go back to that later if there's interest.

But the most significant parameters in that "expansion history equation" are the Hubbletime parameters 13.9 and 16.3 billion years.

THOSE TWO TIME QUANTITIES *SHAPE* THE GROWTH CURVE.

If you change them the scalefactor curve a(t) showing the growth of a generic distance will change.

These two quantities are worth understanding. They express the CURRENT percentage growth rate of distance and the eventual longterm LIMIT growth rate that the present one is slowly tending towards.

I'll pick up and continue from here in my next post. This is enough for now.

 

[fat f...]

nobody replied to me___________________http://www.astro.ucla.edu/~wright/CMB-MN-03/FRL-28Oct08clean.pdf (last pic)

#do people on galaxies which are 13bly away see "first galaxies" when they look towards milky way?#

and do they see that milky way and andromeda are moving away from each other and/or away from observer's direction (like observers on Earth see how galaxies are moving away in the distant space(like in picture))................

here it looks like implosion(inside of a sphere), because dark age can be observed around the universe(in each direction)

and this looks like surface of a sphere model

is reality in between these models? then it would be like a donut model. if there is no centerpoint which serves as center of universe then it would support what i put into "# #"

................this doesn't help stuff because if universe would be this way then there one could also look back(away from the center):
http://en.wikipedia.org/wiki/File:Embedded_LambdaCDM_geometry.png

..............

was explosion(big bang) this enormous that space expands with 70mpc / s? is inflation responsible for this? and is speed the same as it was billions of years ago or does it become faster(expansion)? and does it expand now with speed of 2.16 trilliards(number with 21 characters) km/s?...............does somebody understands it trully? i follow scientifical concept to understand something and if i take away one step before another i can't precede, understanding starts from simple and gets more complicated once something is understood but i read stuff where people start from the middle and then its complicated, then i must ask things(puzzle parts) and if they don't contradict each other, i get the picture from answers

did knowledge came(to cosmologists) from observing mathematical equations which don't contradict each other and was it then applied to understanding(imagination) which then trully understood how universe works/looks like?

 

[twofish-quant]

The are different ways that try to show the same thing.

The first diagram is a two dimensional diagram in *time*. What it shows is what you see if you look back in time.

The second diagram is a three dimensional diagram in both space and time.

To get from the second diagram to the first, imagine a cone that ends at now, and shows the path of light that is arriving at you at this very moment. The intersection between the second diagram and the "light cone" will get you the first diagram.

 

[marcus]

Hi Twofish! Glad to see you! Let's see if we can get F.F. to start his own thread(s) with these questions.He has so much to ask and learn about I fear it would overload this thread.

http://www.astro.ucla.edu/~wright/CMB-MN-03/FRL-28Oct08clean.pdf (last pic)

#do people on galaxies which are 13bly away see "first galaxies" when they look towards milky way?#
...
...

does somebody understands it trully? i follow scientifical concept to understand something and if i take away one step before another i can't precede, understanding starts from simple and gets more complicated once something is understood but i read stuff where people start from the middle and then its complicated, then i must ask things(puzzle parts) and if they don't contradict each other, i get the picture from answers

did knowledge came(to cosmologists) from observing mathematical equations which don't contradict each other and was it then applied to understanding(imagination) which then trully understood how universe works/looks like?

You have many questions--many things that you don't understand and want to talk about.
Too many for this thread. You should start your own thread. Start off with one clear question. Don't ask everything all at once.

Like start a thread with this question (it is a good one)
"#do people on galaxies which are 13bly away see "first galaxies" when they look towards milky way?#"

That is a really good question. If you start a thread with just that, I would certainly answer. Other people would also. You might get several hours of people's time discussing that. Clarifying confusions about how distance is measured in cosmology.

BTW personally I think the first cosmological knowledge did not come from equations.

One of the first bits of knowledge came to a man named Anaxagoras in 250 BC (before there was equation-solving as we know it) by carefully reasoning about the distances to things.. He figured out that the sun is more than 10 times farther than the moon.
This enabled him to deduce that the width of Earth shadow (at the distance of the moon) was nearly as wide as the Earth itself. Then he observed that the Earth shadow when cast on moon during eclipse, was about 3 times greater than the width of the moon.
So he could estimate that the Earth itself (being slightly wider than its shadow at that distance) is slightly more than 3 times wider, maybe something around 4 times wider, than the moon. Figuring that out from scratch is no small achievement!

Since the time of Anaxagoras, most new knowledge about cosmos has evolved by careful reasoning about distances. What Hubble did was not so different from Anaxagoras. He learned a new way to estimate distances to galaxies, and when he surveyed them he found that distances to most galaxies were increasing a certain tiny percentage each year. Discovering expansion this way came, for him, before understanding and believing the Einstein equation--although that equation has geometric expansion as one of its most likely solutions.

From Anaxagoras to Hubble cosmologists have gained knowledge primarily by reasoning carefully about and devising smart ways to estimate the distances to things. So if you want to understand, a good way is to begin at the beginning and ask yourself how do you know that the sun is more distant than the moon.

Or start a thread with that ONE question you asked. Get people to explain the answer to that one question. But not in this thread, it would get too far off the current topic.

 

[twofish-quant]

1) Does somebody understands it trully?

As far as the parts for the diagram that you are showing, people understand it pretty well. Part of the reason is that there is very little "weird physics"

 

[marcus]

About the general question "how do we know", I'll add to what I said a couple of posts back:

From Anaxagoras to Hubble cosmologists have gained knowledge primarily by reasoning carefully about distances and angles (often as seen from another observer's viewpoint) and by devising ways to estimate distances to things.

It's not a bad idea to go over some of the steps in that long human history of accumulated insight, and in effect re-experience. For instance Anaxagoras (the name means "kings market") had the idea to visualize the angle between Earth and sun that someone on moon would see (when we see a half-moon) and realize that it was a right angle.

So he could sketch a right triangle, with the square corner at the moon, and realize that the sun was much farther from us than the moon (because of the near-right angle between them that WE see from earth, at half moon time.)

Now remember from that "much farther" and watching an eclipse he could tell that Earth is something like 4 times wider than moon. But the angle the moon makes in the sky is only 1/120 of a SIXTH of a circle! (Greeks learned from Babs that it's sometimes smart to judge angles as fractions of a SIXTH of a circle rather than of a whole circle.) Which means its distance from us is 120 TIMES ITS WIDTH! That would mean, if Earth is 4 times wider than moon, that distance to moon is 30 Earth diameters.

These complicated chains of reasoning about distances and angles are still at the heart of cosmology. If you practice on Anaxagoras it might make it easier to overcome confusion about the temperature map of the microwave background (the most ancient light we can see.) The angular sizes of its fluctuations are the analogs of the angles and proportions Anaxagoras perceived in the sky.

 

[marcus]

To get back to discussing the simple one-formula model cosmos, here is what I was saying a few posts back:
==================
The main equation in this model is this one. It gives the scalefactor a(t) at each time, going back pretty far into the early history of expansion where it gets a bit off track (because when you get back to the first few million years much of the density in the universe was radiation rather than particles of matter, and radiation behaves differently in expansion , so the physics is not as simple. Anyway the main model equation is this:

\left(\frac{(\frac{16.3}{13.9})^2-1}{(tanh(\frac{1.5}{16.3}t))^{-2}-1}\right)^{1/3}

The coefficient 1.5 and the exponent 1/3 both reflect the fact that we're in a matter dominated era, and have been since the first few million years. Matter density falls off as volume increases---as the cube of distance---and that is where the 3 and the 1.5 come from. In a radiation dominated world they would be 4 and 2. I derived some equations earlier in this thread and can go back to that later if there's interest.

But the most significant parameters in that "expansion history equation" are the Hubbletime parameters 13.9 and 16.3 billion years.

THOSE TWO TIME QUANTITIES *SHAPE* THE GROWTH CURVE.

If you change them the scalefactor curve a(t) showing the growth of a generic distance will change.

These two quantities are worth understanding. They express the CURRENT percentage growth rate of distance and the eventual longterm LIMIT growth rate that the present one is slowly tending towards. Each of the (rather long) intervals of time is the reciprocal of a (rather slow) instantaneous distance expansion rate.

For example the presentday Hubbletime 13.9 billion years can be understood intuitively by thinking that ONE PERCENT of it, namely 139 million years, is the time a distance would take to increase by one percent.
(Continuing steadily at its present speed of growth.) The two hubbletimes 13.9 and 16.3 billion years are convenient to handle and rememeber, and we use them as the two main parameters in the model, but the actual growth rates we are concerned with are their RECIPROCALS which you can write as 1/13.9 per billion years and 1/16.3 per billion years---fractional rates of growth. These are ridiculously (I should say "astronomically") slow rates of fractional growth. So it's easier to work with the times than with the rates.

Going back to the earlier post, here are some sample outputs from our model's main formula: scalefactors a(t) calculated for various times t;
==================
1--- 0.1471433... (when the universe was 1 billion years old, distances were 14.7% what they are now)
2--- 0.2342347... (at 2 billion years, distances were about 23% what they are now)
...
...
13.759--- 0.9999836... (at the present age of 13.759 billion years distances are of course 100% of their present lengths.:-)
...
20--- 1.5235746... (at age 20 billion years, distances will be 52% bigger than they are today.)
==================

What I want to explain is how we can use the output from this one formula to find out other things:
DISTANCES to sources which emitted the light we're getting from them at various times in the past.
DISTANCES to sources whose light comes to us with wavelengths expanded by some factor
SPEEDS that the distances computed as above are now increasing, and were increasing when the light was emitted.

I'll try to get to that in the next post.

 

[marcus]

To begin to deal with distances in terms of this simple model I need to add up this cumulative sum of terms t_s where s is the reciprocal scalefactor 1/a and run thru some range like [1,2] in steps of 0.1, say. Let's start it at s = 1.1 and go 1.2, 1.3,... and at each value of s we will evaluate this formula
atanh((.375136* s^3+1)^-.5)
using http://web2.0calc.com
to get t_s the time (expressed in billions of years) when the light was emitted that we now see wave-lengthened by a factor of s.

1.14678...(starting with 1.1)
2.1893904284755
3.14075638766957
4.01193171154709
4.81239653369235
5.55030772362728
6.2327011396146
6.86566003274671
7.45445670103674
8.00367220130405 (this was for 2.0)
8.51729768096363
8.99882016696824
9.4512951759886
9.87740815857389
10.2795265021777
10.65974356945189
11.0199160299271
11.36169555160973
11.68655575226624
11.99581516649566 (the last one was for 3.0)

So how to use this cumulative sum to get distances? Well suppose a galaxy's light comes in with a scaleup s=2.0 (wavelengths twice as long as when emitted.)
The number from the list we use is the one for 1.9 namely 7.45445670103674, and that gets multiplied by the Δs, the step size, which is 0.1

In addition there are two other things to do: add (1+Δ/2)*t₁
which is 1.05*1.2661864372681=1.329495759131505
and subtract (2-Δ/2)*t₂ = 1.95*0.54921550026731=1.0709702255212545
The difference is 0.2585255336102505, so that's what gets added:
0.745445670103674+0.2585255336102505 = 1.0039712037139245
Finally multiply that by 16.3/1.5 and get 10.9098... Gly.
Ned Wright's calculator says 10.901 Gly (with equivalent model parameters). So the accuracy is not so bad.
that's the current distance: 10.91 billion lightyears.

All those extra decimals are ridiculous but it is too much trouble to be rounding off all the time so I just take what the calculator gives and finally round it off to something sensible at the end.

the corresponding thing for s=3
1.168655575226624+1.329495759131505-0.91231527197679=1.585836062381339
and then again finally multiply by 16.3/1.5, to get 17.23275...billion lightyears
Ned Wright says 17.220, so again we are off by 1 in the fourth digit.

 

[marcus]

I didn't get around to editing the previous post until after the deadline for changes and it needs some clarification.
The key formula in the toy version cosmic model I'm working with is
(16.3/1.5)*atanh((.375136* s^3+1)^-.5)
This gives the time (expansion age in billions of years) at which the reciprocal scalefactor was a particular value s.
Another way to say it: t_s is the time (expressed in billions of years) when the light was emitted that we now see wave-lengthened by a factor of s.
s can be thought of, if you like, as the "scale-up factor": because since the time t_s, distances and wavelengths have been scaled up by a factor of s. Back at time t₂ distances were 1/2 their present size, back at t₉ distances were 1/9 their present size, t₁ is the universe's present expansion age, and so on. I won't be using the traditional notation for s, which is "1+z", since it makes the formulas even messier than they are already.

The Hubbletime parameter 16.3 billion years represents the cosmological constant (asymptotic distance growth rate.) The other Hubbletime parameter, giving present growth rate, is 13.9 Gy and since that most often appears in combination as (16.3/13.9)²-1=0.375136... I have, for simplicity, packaged it in that number.

When I have to do a lot of calculating, e.g. numerical integration, I leave off the coefficient (16.3/1.5) and factor it in only at the end.

It turns out that we can do a pretty good job of estimating the distances to sources at various scaleups by essentially just adding up a long string of arctanh values, with s advancing from 1 to s in steps of some smallish stepsize Δ. If we take Δ = 0.1, this amounts to:

1.05t₁+ 0.1( t_1.1+ t_1.2+...+ t_s-0.1) - (s-0.05)t_s

This is the bare bones of a numerical integration for c∫ s dt.
The idea is that at each interval dt of time in the past, the light from an object travels a distance cdt and this gets scaled up by a the appropriate factor s. So at present the distance to the object is the sum of all those scaled-up segments and equals c∫ s dt.

It looks messy but seems to work out all right.
Here's a cumulative sum of atanh((.375136* s^3+1)^-.5)
using http://web2.0calc.com

1.14678...(starting with 1.1)
2.1893904284755
3.14075638766957
4.01193171154709
4.81239653369235
5.55030772362728
6.2327011396146
6.86566003274671
7.45445670103674
8.00367220130405 (this was for 2.0)
8.51729768096363
8.99882016696824
9.4512951759886
9.87740815857389
10.2795265021777
10.65974356945189
11.0199160299271
11.36169555160973
11.68655575226624
11.99581516649566 (this was for 3.0)
12.29065686210409
12.57214523554962
12.84124042899438
13.09881073851359
13.34564332215593
13.58245346592667
13.80989262370986
14.02855541222707
14.23898571313266
14.44168201025018 (for 4.0)
14.63710206991032
14.82566705565418
15.00776515463345
15.18375478139281
15.3539674149637
15.51871011700947
15.67826777187171
15.83290508355766
15.98286835979816
16.12838710914517 (for 5.0)

So suppose we evaluate 1.05t₁+ 0.1( t_1.1+ t_1.2+...+ t_s-0.1) - (s-0.05)t_s
to find the distance now
to a source with s = 4 (its light comes in wavelengthened by a factor of 4).
As long as we are using the stepsize Δ=0.1, the first terms is always 1.329495759131505,
and the sum, multiplied by the stepsize, can be read off that list: 1.423898571313266.
The term at the end, that gets subtracted, is 3.95*0.202696297=0.800650373

1.329495759 +1.423898571 - 0.800650373 = 1.95274396

And then at the end the whole thing gets multiplied by the cosmological constant term 16.3/1.5,
to give 21.2198≈ 21.22 billion light years.
Sorry about all the meaningless extra digits but it is too much trouble to be rounding off every time I take a result from the calculator, so I just round off at the end.
Let's compare this with Wright's calculator.
Well, Wright's says 21.204 Gly. So as usual we are OK for three significant figures and off in the 4th place.

Notice that since this numerical summing procedure gives us the NOW distance to the source, all we need to do is multiply by the scalefactor a, or alternatively divide by the scaleup s, and we get the THEN distance---how far from our matter or galaxy the thing was when it emitted the light.

So this primitive model already does quite a bit that one expects from serious cosmology calculators. Given a scalefactor a (or the reciprocal 1/a = s) it can give the corresponding expansion age---the time when the source galaxy emitted the light. And it can give the Now and Then distances to the source (proper distance, as if you could halt expansion at the given moment and measure directly).

The THEN distance is essentially the ANGULAR SIZE distance (our model is spatial flat) so that's taken care of.

It still might be nice to be able to calculate the HUBBLETIME corresponding to a given scalefactor a, or its reciprocal s. That is hour handle on the rate of expansion going on at the time the light was emitted.

Notice that the light itself, when it arrives, tells us the scalefactor, or equivalently its reciprocal s, which we focus on here, so the other things we want to know should be calculated from s.

 

[marcus]

It looks as if the Hubbletime Y_s corresponding to a given scaleup s should be given in billions of years by:

16.3(0.375136s³+1)^-.5=16.3/sqrt(.375136*s^3 + 1)

so using the calculator let's try that for s=3

It gives Y₃ = 4.8861403 ≈ 4.886 billion years.

Let's see if I've made a mistake.

Apparently not, Jorrie's calculator (with the corresponding parameters) gives 4.885 billion years.


Remember that the two parameters we're using in the model's formulas, namely 16.3 and 0.375136, are just an equivalent form of the two Hubbletimes which determine two key expansion rates, now and in distant future.
Y_now = 13.9 billion years

Y_∞ = 16.3 billion years

The number .375136 is simply what (Y_∞/Y_now)² - 1 = (16.3/13.9)² - 1 works out to be.
You can think of 1.375136 as a ratio of two expansion rates, squared. It is simply (H_now/H_∞)² so it tells you how much more the percentagewise expansion rate is now than it will be in the longterm future.

When people talk about "acceleration" what they mean is what you see when the H expansion rate is declining only very slowly or is steady at some given value. As long as H is not declining too rapidly, if you watch a particular distance it will grow by increasing annual amounts as the principal grows. Not terribly dramatic, given the very low "interest rate" but there is acceleration in a literal sense.

In the previous post we calculated that a galaxy we see with scaleup factor s = 4 (wavelengths quadrupled) is now at a distance of 21.22 Gly.
How fast is that distance now growing?
That is very simple to calculate. We just divide 21.22 Gly by 13.9 Gy.
D_now/Y_now = 21.22/13.9 = 1.53c.
That means it is growing at 1.53 times the speed of light. Calculation easy with these quantities

For comparison and a bit more practice, back in post #433 we found that the distance NOW for s=3 was 17.23275 billion lightyears, which means distance THEN was 17.23275/3=5.744 billion lightyears.

However we just found that also for s=3 we have Hubbletime Y_then = 4.886 Gy.

So for a s=3 galaxy, whose distance THEN at time light was emitted was 5.744 Gly, how fast was that distance then growing?
Well obviously D_then/Y_then = 5.744 Gly/4.886 Gy = 1.18 ly/y = 1.18c.

The notation is still far from perfect, but I hope some of this is comprehensible
==============
referring back to post #434 the last increment was 0.14551874934701
and to find the now distance to an s=5 source D₅(now) one would take 4.95*0.14551874934701=0.720317809 off at the end, so it looks like
1.329495759 +1.598286836 - 0.720317809 = 2.207464786
which then gets multiplied finally by 16.3/1.5 to give 23.987784 billion lightyears.
So unless I've made a mistake that's the distance now to an s=5 source. I'll compare with what Wright's says.
It says 23.970 Gly. So we are still OK for three digits.
D₅(now) = 23.99 Gly
D₅(then) = 23.987784/5 ≈ 4.798 Gly
Y₁ = 13.9 Gy
Y₅ = 16.3/sqrt(.375136*5^3 + 1)= 2.35535 ≈ 2.355 Gy

So we can say that for an s=5 galaxy, when the light we are now getting was emitted, the distance was expanding at a speed 4.798 Gly/2.355 Gy = 2.037 c, over twice the speed of light. The then and now speeds of expansion are given by:
D₅(then)/Y₅ and
D₅(now)/Y₁

 

[Jorrie]

It looks as if the Hubbletime Y_s corresponding to a given scaleup s should be given in billions of years by:

16.3(0.375136s³+1)^-.5=16.3/sqrt(.375136*s^3 + 1)

It appears to me that your 'scaleup factor' (s = 1/a = z+1) is a useful one, since it does not go negative for future times, but just goes smaller than 1. I would caution against the terminology though, as it is too close to the conventional 'scalefactor' and may cause confusion. Maybe something like 'upscale ratio' or 'expansion ratio'? I would prefer 'upscale ratio of distances' rather than 'scaleup factor of wavelengths', so as to not also cause potential confusion with Doppler effects.

I'm working on a variant of the cosmo-calculator that will give a table for a range of z (or s?), with some useful values in the columns. Not quite there yet, but it looks practical.

 

[marcus]

How about calling it "stretch factor"?
Or "extension ratio"?
Another idea, similar to something you suggested is to call the s number the "enlargement"
because it is the ratio by which distances are enlarged during the time the light is on its way
somewhat reminiscent of photographs being blown up.
I'd welcome more suggestions.
I'm glad to hear your new calculator is looking practical! I think the idea of generating tables for a range of z (or for s !) is a good one. Even fairly short tables with 5 to 10 lines can give someone extra perspective and intuition about how things are evolving. Just being able to compare two or three lines can be informative. After trying different things I do agree that the reciprocal scalefactor (whatever you call it and whether or not you subtract 1 from it) is the most useful handle on the situation.

 

[marcus]

Here's one way to think about it: say we number the stages of expansion history according to how much distances have been enlarged since then.
It means that earlier slices of spacetime have larger s numbers, which at first seems turned around, but in fact it's cleaner formula-wise to do the numbering backwards that way.

To illustrate, suppose we are watching a galaxy as it was when distances were 1/4 of present size. We can denote that stage of expansion history by saying s = 4. While the light was on its way, distances and wavelengths have been enlarged by that factor. So that galaxy, as we see it, is in "slice 4" of expansion history.

In that way of denoting stages of expansion, the present is s=1, because enlarging by a factor of 1 is the identity.

In the simplified toy model, the expansion age t_s associated with a stage s is given (in billions of years) by:
t_s = \frac{16.3}{1.5}arctanh \left( \left( 0.375136 s^3 + 1\right)^{-1/2}\right)
And the corresponding Hubbletime at stage s, also in billions of years, is:
Y_s = 16.3 \left( 0.375136 s^3 + 1\right)^{-1/2}
These are the two basic equations of the model--the other usual quantities such as distances and expansion speeds can be derived from these two. There is one peculiar thing to notice, which is that with expansion stages numbered this way, not only do we have the present tagged s = 1 but also future infinity is s = 0.
So the eventual, or longterm value of the Hubbletime (a key parameter in the model) is:
Y_0 = 16.3 \left( 0.375136 \times 0^3 + 1\right)^{-1/2} = 16.3 Gy
while the present Hubbletime is:
Y_1 = 16.3 \left( 0.375136 \times 1^3 + 1\right)^{-1/2} = 13.9 Gy
I keep having to write this number 0.375136, which is kind of like a parameter of the system being the square ratio of our two Hubbletimes, less one. (Y₀/Y₁)² - 1.
So I will call that number capital Theta Θ. The two basic equations of the model are then:

t_s = \frac{2}{3}Y_0 arctanh \left( \left( \Theta s^3 + 1\right)^{-1/2}\right)
Y_s = Y_0 \left( \Theta s^3 + 1\right)^{-1/2}
People who don't like greek letters should just remember it is a shorthand for an ordinary number ≈ 0.375 that essentially says something about the amount bigger current expansion rate is than the eventual longterm rate. (their ratio is about sqrt(1.375)

 

[marcus]

I want to try out some terminology in part suggested to me by Jorrie's comments and which he might be puttng to use in another project.But I can try several ideas out, tentatively, in connection with this simple cosmic model. The main variable could be called the "stretch" because it is the factor by which distances from a past slice of spacetime are enlarged (and wavelengths too) between then and now.

The idea is that if you start back to some earlier stage in expansion history and the enlargement of distances (and wavelengths) from then to now is a stretch factor of four (say S = 4)
then the scale back then, relative to now is 1/4, or 0.25.
So the stetch and scalefactor are reciprocals, like 4 and 1/4.

It just turns out that the stretch is a convenient variable to run the model, or the calculator, on. You get the simplest formulas that way, of the various things I've tried.

So I'm using S to stand for the stretch and the conventional letter a (= 1/S) to stand for the scale factor. The lineup of numerical information could (tentatively) go like this:
Stretch---Scale factor---Expansion age---Hubble time---Distance now---Distance then

and then again, this time showing the symbols that might be used to denote these quantities:

Stretch (S=1/a)---Scale factor (a)---Expansion age (t_S)---Hubble time (Y_S)---Distance now (D_S[now])---Distance then (D_S[then])

The idea is, we observe a galaxy and its light tells us the stretch factor S, say it is 4. Wavelengths 4 times what they were at the start of the trip. The galaxy is living back when distances were 1/4 present size. Then D₄[now] tells us proper distance to the galaxy NOW, and D₄[then] tells us distance back then, when light was emitted, from our matter (that became us) to the galaxy.

If we want to know the SPEED of distance growth, you simply divide the distance by the Hubble time belonging to that slice. Back then when the distance was D₄[then], it was growing at speed D₄[then]/Y₄.

The present is denoted S=1 and the present Hubbletime is Y₁ = 13.9 billion years. So the present distance is expanding at speed D₄[now]/Y₁.
Today's distances, if you want to know what speed they are expanding, you just divide them by 13.9 billion years.

So that's a provisional idea for a list of 6 related numbers that the model, or a calculator, can give you, that seems like enough to work with and get a picture of the expansion history from.

 

[marcus]

I don't want to forget that simple model, although Jorrie has now put an excellent online tabulator on line.
Here is the single-line formula calculator we were using, see post #434
http://web2.0calc.com

Here is the calculator formula to compute the time given the stretch S:
(16.3/1.5)atanh((.375136* S^3+1)^-.5)

There's also a more complicated version for it given the scalefactor, but we probably won't use it.
(16.3/1.5)atanh((((16.3/13.9)^2 -1)/a^3+1)^-.5)

I just had a kind of exciting look into the future. I went to web2.0calc and put in exactly what I mentioned, namely
(16.3/1.5)atanh((.375136* S^3+1)^-.5)

And decided to see when distances would be 100 times what they are today
which means scalefactor a=100 and reciprocal S = 1/a = 0.1

So I put 0.1 in place of S, in the formula and pressed = and it said that would happen
in year 87.92 billion.

So that is kind of cool. When expansion has been going on for about 88 billion years distances can be expected to be about 100 times what they are today.

Let's try another. when will it be that distances are FIFTY times what they are today? Put in S=0.02 to the web2.0calc.
Bingo. It says that will happen in year 76.625 billion.

And just as a side comment with continuous compounding the Hubbletime 16.3 billion years corresponds to a doubling time of 11.3 billion years. That is the natural log(2) times 16.3. So it seems right that you go from scale 50 to scale 100 in something a little over 11 billion years.

 

[marcus]

In post#434 I used a crude numerical integration of Sdt to find the distance now to a galaxy in the past in era S
It turned out that when you rearrange an Sdt integration to make it easy to add up you get what LOOKS like a tdS integration (with extra terms at either end). this is just algebraic rearrangement. Then the steps can be of S rather than time and we can use the formula t_S
(16.3/1.5)*atanh((.375136* S^3+1)^-.5)
which gives the time (expansion age in billions of years) when the reciprocal scalefactor (stretch) was a particular value S

In the earlier post we had S advancing from 1 to S in steps of some smallish stepsize Δ. If we take Δ = 0.1, this amounts to:

1.05t₁+ 0.1( t_1.1+ t_1.2+...+ t_s-0.1) - (s-0.05)t_s

This is what the numerical integration for c∫ S dt boiled down to.
The idea was that at each interval dt of time in the past, the light from an object travels a distance cdt and this gets scaled up by the appropriate factor S. So at present the distance to the object is the sum of all those scaled-up segments and equals c∫ S dt.
====================

So I decided to look into the future with the same technique and I found that if a galaxy is going to pass thru your forward lightcone at S=0.5, that is when distances are TWICE what they are today, then the distance NOW to it is 7.5 Gly.
Where are the galaxies NOW which you could hit with a flash of light you send today and which arrives wavestretched to double length? They are 7.5 billion lightyears from here.
It's like a time reverse image of the earlier game when we asked things about a galaxy whose light comes in wavestretched by a factor of 2, where was it when it emitted the light, where is it now etc.

The numerical integration boiled down to:
.55t_.5+ 0.1( t_.6+ t_.7+...+ t_.9) - .95t_1.0

And I evaluated that and got 7.5 billion lightyears.

So then we can say that when the signal we send arrives the distance to the target galaxy will be 15.0 billion light years. Because we know the expansion of scale between now and that time in the future.

 

[marcus]

In the previous post I did a rough numerical integration based on toy model and got the estimate that a galaxy we can send a message to which will arrive when distances are TWICE today (namely an S=0.5 galaxy) is currently at distance 7.5 Gly and when the message arrives it will be at S=15 Gly.

Now I can confirm that with the A20 calculator, which sees the shape of expansion history in the future as well as the past
http://www.einsteins-theory-of-relativity-4engineers.com/CosmoLean_A20.html

I just make a small table running from present S=1 out to S=.1 in future, in steps of 0.1.

So it covers the S=0.5 case but also gives me a little context (to help grow intuition/feel for the expansion process.)

===quote===
Hubble time now (Ynow) 13.9 Gy Change as desired (9 to 16 Gy)
Hubble time at infinity (Yinf) 16.3 Gy Change as desired (larger than Ynow)
Radiation and matter crossover (S_eq) 3350 Radiation influence (inverse: larger means less influence)
Upper limit of Stretch range (S_upper) 1.0 S value at the top row of the table (equal or larger than 1)
Lower limit of Stretch range (S_lower) 0.1 S value at the bottom row of table (S_lower smaller than S_upper)
Step size (S_step) 0.1 Step size for output display (equal or larger than 0.01)

Stretch (S) Scale (a) Time (Gy) T_Hubble (Gy) D_now (Gly) D_then (Gly)
1.000 1.000 13.756 13.900 0.000 0.000
0.900 1.111 15.250 14.444 -1.417 -1.575
0.800 1.250 16.981 14.929 -2.887 -3.608
0.700 1.429 19.004 15.342 -4.401 -6.287
0.600 1.667 21.396 15.677 -5.952 -9.921
0.500 2.000 24.279 15.930 -7.533 -15.066
0.400 2.500 27.856 16.108 -9.135 -22.839
0.300 3.333 32.507 16.218 -10.752 -35.840
0.200 5.000 39.097 16.275 -12.377 -61.886
0.100 10.000 50.388 16.297 -14.006 -140.059
===endquote===

So the quick and dirty estimate I did earlier worked OK. For an S=.5 galaxy (where our message reaches when distances are TWICE) the present distance really is 7.5 and the distance then when message arrives really is 15 Gly.

the minus signs have to do with the direction the light is going, from us to them.
whereas in the past the distances have positive sign because the light is coming from them to us----itself a kind of nice feature.

Also as an extra bonus the A20 tells me that the message that we send today (expansion age 13.75 billion years) will arrive when expansion age is 24.3 billion years. So it will take around 11 billion years to get there.

That makes sense: when it arrives at destination the message will be 15 billion lightyears from us, and will have been traveling 11 billion years---you have to allow for some expansion of distances so naturally 15 > 11.

 

[marcus]

The last 5 pages or so have been largely devoted to discussing the development of what I think is a really fine cosmology teaching/learning calculator, by Jorrie, and also working out some simplified equations that approximately reproduce part the expansion history (after the radiation energy density stopped being a major factor in the early U.) I should stress that Jorrie's calculator is what I would call professional grade--it reproduces the standard cosmic model--whereas the other thing we were working on is different, more of a "toy model".

Now the calculator (currently version A25) has its own sticky thread "Look 88 billion years into the future..." and I'd like to find a way to get this thread back into the groove of helping to "get us all on the same page."

One thing that could be highlighted, that we haven't discussed much here so far, is that if a massive particle or object is given a kick so that it has its own individual motion relative to the universe rest frame---the "Hubble flow"---CMB rest, it will gradually slow down relative to CMB rest and given enough time will REJOIN the "Hubble flow", or come approximately to a STOP relative to the ancient light.

This is rather un-Newtonian and could be unintuitive to newcomers. It violates conventional conservation notions. But it is really basic to understanding so we should talk about it. It is analogous to the redshifting of light. the light loses energy and momentum as it travels across cosmological distances, although its speed doesn't change.
With a massive object, the mass doesn't change but the speed does, so there is the same loss of energy and momentum.

A thing's individual velocity relative to the ancient light is called its peculiar velocity (meaning "special to itself", not weird). The basic message is if a massive object is given a kick so it acquires some peculiar velocity (relative to CMB rest) then over a long period of time that velocity will tail off to zero and it will asymptotically come to rest.

This figures in discussions of the "tethered galaxy problem". We were discussing that in another thread and Jorrie plotted some informative curves . For now, at least, I'll just give a link to his post:
https://www.physicsforums.com/showthread.php?p=4082096#post4082096

BTW when thinking about the balloon analogy it's good to remember that fixed points on the balloon surface represent points at CMB rest. A galaxy at some fixed point like that sees the CMB the same temperature in all directions, instead of having a doppler hotspot caused by its own peculiar motion. In Ned Wright's balloon animation all the galaxies are at rest (no peculiar motion) shown by their staying always at the same latitude and longitude on the balloon surface. The photons on the other hand have motion. Each moves at constant speed in constant great-circle direction. However if you watch carefully will see their wavelengths enlarge to symbolically show redshift.

 

[Jorrie]

One thing that could be highlighted, that we haven't discussed much here so far, is that if a massive particle or object is given a kick so that it has its own individual motion relative to the universe rest frame---the "Hubble flow"---CMB rest, it will gradually slow down relative to CMB rest and given enough time will REJOIN the "Hubble flow", or come approximately to a STOP relative to the ancient light.

This is rather un-Newtonian and could be unintuitive to newcomers. It violates conventional conservation notions. But it is really basic to understanding so we should talk about it. It is analogous to the redshifting of light. the light loses energy and momentum as it travels across cosmological distances, although its speed doesn't change.
With a massive object, the mass doesn't change but the speed does, so there is the same loss of energy and momentum.

Actually, there is a way in which the balloon analogy can make cosmic particle momentum decay intuitive. Simply consider a massive, frictionless particle that moves along the surface of the spherical balloon as a Kepler orbit around the center of a balloon. This particle must conserve angular momentum relative to the center of the balloon, i.e.

L = r^2 m d\phi/dt = constant. Since v = r d\phi/dt, it means that for non-relativistic speeds, the particle speed scales with 1/r.

If the balloon is being inflated, the particle must lose surface speed, just like a Kepler orbit that is losing orbital speed at larger radius. If the increase in balloon radius is kept up, the particle’s surface speed will eventually approach zero, as radius tends to infinity. In cosmology, this is usually described as 'joining the Hubble flow'.

The analogy seems to hold even for relativist particles. The relativistic Keplerian equation for the conservation of orbital angular momentum is:

L = (1-v^2/c^2)^{-0.5} r^2 m d\phi/dt = constant (e.g. MTW eq. 25.18).

This simple scheme can be shown to reproduce the curves of figure 3.5 obtained by Davis (2004) [http://arxiv.org/abs/astro-ph/0402278] (with the exception of v=c).

I think that the v=c case can also be handled by the analogy; the particle’s momentum must then be expressed in terms of the de Broglie wavelength.

Edit:
Relativistic de Broglie wavelength is given by: \lambda=\gamma h/(m v), where \gamma is the Lorentz factor.

If we write the angular momentum of the 'balloon particle' in terms of surface velocity, it is simply L = \gamma r m v. Taking \gamma from the de Broglie wavelength, gives the conservation of angular momentum as

L = r h / λ = constant, valid for photons and matter.

 

[PeterJ]

I have no ability to understand a sphere with nothing off the surface. I'd rather have Plotinus's hypersphere. This is more or less a balloon analogy, with the spacetime universe on the surface as usual, but it has a centre and at least one extra dimension. Then we have the ability to include an extra dimension (or bundle of them) at each point in spacetime, since this would be represented as a connection to the centre point. Plotinus was not talking physics exactly, but his model seems more sophisticated than a 3D balloon representing a 2D spacetime.

But I am way out my depths here.

 

[marcus]

I have no ability to understand a sphere with nothing off the surface. I'd rather have Plotinus's hypersphere. This is more or less a balloon analogy, with the spacetime universe on the surface as usual, but it has a centre and at least one extra dimension. Then we have the ability to include an extra dimension (or bundle of them) at each point in spacetime, since this would be represented as a connection to the centre point. Plotinus was not talking physics exactly, but his model seems more sophisticated than a 3D balloon representing a 2D spacetime.

But I am way out my depths here.

Instructive example! A mystic will postulate additional details (like the "edge" of the universe, or extra spatial dimensions) because they appeal to his imagination. Or even that they are "required" by his imagination.

the type of person we could call pragmatic or perhaps "Occamite" will avoid adding features which lack an operational meaning---i.e. some way to experience, even if very tenuous or indirect.

I would say try to think of the EXPERIENCE of being 2D and living in a 2D sphere. Don't picture the sphere as if you are a God, outside and looking from outside at the sphere. Using some new type of lightrays that travel in 3D rather than 2D. Think of a sphere as the experience of living in it. And also think of a hypersphere that way.

Let's say that you and your brothers discover a remarkable fact about the world namely that there is a special area K which you have determined experimentally which allows you to reliably predict the area of any triangle!
You just have to sum the angles, subtract π, and multiply by that area K!
this always turns out to give the area (if you take the trouble to measure the area carefully.

The rule used by Euclid, namely 1/2 the base times the height does not work for you, it is only approximately right for small area triangles and gets progressively wronger for larger ones.

That's part of what I mean by the experience. It would apply also to living in a hypersphere. It does not involve postulating an extra dimension which we don't experience and cannot access. It just involves experimenting with triangles and determining the value of the area K.

Circumnavigating is another aspect of the experience which you (as creature living in sphere or hypersphere) might have. You can think of various others.

 

[PeterJ]

No it's okay. I'm happy with my way of thinking about it. I'd say that the topography of the universe is unrepresentable as a visual model, and so within quite wide limits it would be a matter of personal preference how we do it. A Klein bottle or Necker cube would also be relevant images.

I'm not sure what you mean about 'mystics' and the stuff that appeals to their imagination. It has nothing to do with imagination. If Plotinus is to believed he is trying to describe what he is seeing, and he was not seeing any edges, nor any inside or outside. Perhaps he is not to be believed, but his model does at least allow for the idea that distance is arbitrary, which seems to make it useful, and it's only one more dimension, making it more economical than string theory. He even adds the proviso 'it is as if'.

I'm not suggesting that his description is 'true', just something to consider.

 

[marcus]

New narrowed-down values of the cosmological parameters, coming out of the SPT (south pole telescope)
http://arxiv.org/pdf/1210.7231v1.pdf
Scroll to Table 3 on page 12 and look at the rightmost column which combines the most data:

Ω[SUB]Λ[/SUB]     0.7152 ± 0.0098
H[SUB]0[/SUB]     69.62 ± 0.79
σ[SUB]8 [/SUB]    0.823 ± 0.015
z[SUB]EQ[/SUB]    3301 ± 47

Perhaps the most remarkable thing is the tilt towards positive overall curvature, corresponding to a negative value of Ω_k

For that, see equation (21) on page 14
Ω_k =−0.0059±0.0040.
Basically they are saying that with high probability you are looking at a spatial finite slight positive curvature. The flattest it could be IOW is 0.0019, with
Ω_total = 1.0019
And a radius of curvature 14/sqrt(.0019) ≈ 320 billion LY.
Plus they are saying Omega total COULD be as high as 1.0099 which would mean
radius of curvature 14/sqrt(.0099) ≈ 140 billion LY.

So the idea which is traditionally favored of perfect flatness and spatial infinite is hanging on by its 2 sigma fingernails. It is still "consistent" with the data at a 2 sigma level.

But the hypersphere ( abbreviated S³ for the 3D analog of the 2D surface of a ball ) is looming realer and realer as a kind of ignored elephant in the room. It could still go away of course. We avert our eyes and hope it will have the politeness to do so.

For Jorrie's A25 calculator the important parameters as estimated by the SPT report are
current Hubble time = 14.0 billion years
future Hubble time = 16.6 billion years
matter radiation balance S_eq = 3300

or with more precision put these into Google calculator:
1/(69.62 km/s per Mpc)
1/(69.62 km/s per Mpc)/.7152^.5

 

[hagendaz]

Can I go light speed riding the expansion of space?

On the subject of space expansion...
If I built a spaceship with a machine (an engine if you will) on the tail end of the ship that eliminated all effects of gravity at the tail end of the ship , so as to counter act any gravitational pull on the tail end of the ship, would my spaceship thus be capable of traveling at light speed as I ride the expansion of space in what ever direction my ship is pointed?
Or can I only travel away from the center of the universe?

Would I instantly be moving at the same speed as the space is expanding the moment I turn on my spaceship, but feel no acceleration?

 

[marcus]

On the subject of space expansion...
If I built a spaceship with a machine (an engine if you will) on the tail end of the ship that eliminated all effects of gravity at the tail end of the ship , so as to counter act any gravitational pull on the tail end of the ship, would my spaceship thus be capable of traveling at light speed as I ride the expansion of space in what ever direction my ship is pointed?
Or can I only travel away from the center of the universe?

Would I instantly be moving at the same speed as the space is expanding the moment I turn on my spaceship, but feel no acceleration?

The expansion of distances doesn't GO anywhere. No person or object approaches any destination. Simply put: Things that aren't held together by their own gravity or molecular forces just get farther apart.

There is no "center" that anyone can point to.

So you cannot "ride" the expansion of space in any direction. Since there is no center you cannot " travel away from the center" as you say, either.

Typical very largescale distances grow several times faster than the speed of light but nothing travels anywhere.

The balloon analogy is intended to illustrate those things to make them easy to visualize. You might try studying the brief animation movie of it, reading some of this thread, or the FAQs.

You could start your own thread with this question, since it does not fit in so well in this balloon analogy thread.