Steps on the way to Lightcone cosmological calculator

[marcus]

The new A20 tabular calculator let's you look at changing geometry out to about 88 billion years according to the standard LCDM cosmic model (with usual estimates for the parameters.).
http://www.einsteins-theory-of-relativity-4engineers.com/CosmoLean_A20.html

It's pretty neat. Here is one sample tabulation. Red stuff is just the three standard parameters, estimated based on observation. No reason to change them, although in this calculator you CAN change them and play around to see the effects.
The blue stuff is what I put into give bounds and step size for the table I wanted it to generate
From the present (S=1) to the distant future (S=0.01) when distances are 100 times what they are today. In steps of ΔS = 0.09. those are just what I chose. If you choose a smaller step size like ΔS = 0.01 you get a table with more rows, like around 100 rows instead of only 12 rows. I won't bother to align the columns. It's probably legible as is.
===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) 0.01 S value at the bottom row of table (S_lower smaller than S_upper)
Step size (S_step) 0.09 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 13.769 13.896 0.000 0.000
0.910 1.099 15.104 14.387 -1.219 -1.339
0.820 1.220 16.630 14.829 -2.536 -3.093
0.730 1.370 18.374 15.221 -3.884 -5.320
0.640 1.563 20.402 15.545 -5.270 -8.234
0.550 1.818 22.772 15.812 -6.676 -12.138
0.460 2.174 25.618 16.006 -8.108 -17.627
0.370 2.703 29.120 16.143 -9.555 -25.825
0.280 3.571 33.629 16.233 -11.010 -39.323
0.190 5.263 39.934 16.278 -12.474 -65.650
0.100 10.000 50.390 16.296 -13.939 -139.393
0.010 100.000 87.919 16.300 -15.406 -1540.607

For the model used, see this thread on Physicsforums.
=====endqquote=====

what this tells you, among other things, is which of the galaxies out there you can reach if you flash a signal to them today.

It says ANYTHING THAT IS TODAY NEARER THAN 15.4 BILLION LY is a target you can reach if you flash a message today, and it will get there WITHIN 88 BILLION YEARS.

It also says that 88 billion years from now is when distances will be 100 times what they are today (cosmological distances, not dimensions of bound structures like a rock or solar system)

So if you select a galaxy which is today 15.4 billion LY and you flash a message today, when the message finally gets there the distance to the galaxy (and the message arriving at it) will be 1540 billion LY.
You can read that off the table too.

Is there anyone to whom this does NOT make sense. This is a great calculator and an interactive version of the standard cosmic model that is in professional use (LCDM) and there must be plenty of people who can explain if you find anything obscure about the table. Everybody should get so they understand the table outputs of this calculator both of past history and of the future, IMHO. They are basic.

 

[mfb]

It says ANYTHING THAT IS TODAY NEARER THAN 15.4 BILLION LY is a target you can reach if you flash a message today, and it will get there WITHIN 88 BILLION YEARS.

And anything beyond ~16 billion ly is unreachable?
If I understand this correctly, the range we can contact at all shrinks with ~1 ly / year (using the current distance). This reduces the reachable volume by ~3*10^21 ly^3 per year, more than the volume of our local group (according to WolframAlpha). Claustrophobia anyone? ;)

 

[marcus]

And anything beyond ~16 billion ly is unreachable?
If I understand this correctly, the range we can contact at all shrinks with ~1 ly / year (using the current distance). This reduces the reachable volume by ~3*10^21 ly^3 per year, more than the volume of our local group (according to WolframAlpha). Claustrophobia anyone? ;)

Hi Mfb, I was glad to see your comment! In terms of what is called proper distance the reachable volume is growing. Its current radius is about 15.6 billion ly.
Its radius is expected to plateau at 16.3 billion ly.
So the reachable volume (just the volume of sphere with that radius) is growing and will plateau accordingly.

Note that the CEH is different from the Hubble radius. The Hubble radius is the distance that is growing at rate c. It is currently 13.9 Gly and the CEH (the reachable radius) is 15.6 Gly.
I think you know this but I'll say it just in case others read this.

There is the tricky idea of COMOVING distance, where everything and every galaxy is permanently assigned its present distance and keeps that like a tattoo for all its past and future life.
Comoving volume corresponds intuitively to "amount of matter".
Now because of expansion the number of galaxies within our CEH range---our 15.6 or eventual 16.3---is declining. If you keep a volume at a fixed proper distance size then stuff will leak out of it. So the amount of matter in our reachable sphere-shaped volume is declining. Even though in proper distance terms the volume is not.

So the reachable "comoving volume" (essentially referring to amount of reachable matter) is slated to decline almost to zero. Just an agglomeration of Milky and Andromeda surrounded by a big 16 Gly radius ball with not much in it.

 

[mfb]

If you keep a volume at a fixed proper distance size then stuff will leak out of it. So the amount of matter in our reachable sphere-shaped volume is declining.

That was my point. There are co-moving objects reachable today, but not tomorrow.

 

[marcus]

To see the Cosmic Event Horizon actually emerge in the output of this calculator we have it make a longer table with small step. What I showed in post#1 was a shortened form:
http://www.einsteins-theory-of-relativity-4engineers.com/CosmoLean_A20.html

===quote===
... Red stuff is just the three standard parameters, estimated based on observation. No reason to change them, although in this calculator you CAN change them and play around to see the effects.
The blue stuff is what I put into give bounds and step size for the table I wanted it to generate
From the present (S=1) to the distant future (S=0.01) when distances are 100 times what they are today. In steps of ΔS = 0.09. those are just what I chose. If you choose a smaller step size like ΔS = 0.01 you get a table with more rows, like around 100 rows instead of only 12 rows...


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) 0.01 S value at the bottom row of table (S_lower smaller than S_upper)
Step size (S_step) 0.09 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 13.769 13.896 0.000 0.000
0.910 1.099 15.104 14.387 -1.219 -1.339
0.820 1.220 16.630 14.829 -2.536 -3.093
0.730 1.370 18.374 15.221 -3.884 -5.320
0.640 1.563 20.402 15.545 -5.270 -8.234
0.550 1.818 22.772 15.812 -6.676 -12.138
0.460 2.174 25.618 16.006 -8.108 -17.627
0.370 2.703 29.120 16.143 -9.555 -25.825
0.280 3.571 33.629 16.233 -11.010 -39.323
0.190 5.263 39.934 16.278 -12.474 -65.650
0.100 10.000 50.390 16.296 -13.939 -139.393
0.010 100.000 87.919 16.300 -15.406 -1540.607

===endquote===

One thing this tells us is that if we send a message to a galaxy expecting it to arrive when distances are 100 times what they are now then the galaxy has to be only 15.4 billion ly away.

What is happening is that this distance is CONVERGING to about 15.6 billion ly, as we allow time to run to infinity and the expansion factor 100 to grow without bound.
I have to go out for the evening so can't copy in the table. But try making the table yourself.
Put in
upper bound S = 1
lower bound S = 0.01
step size = 0.01

You will see the NOW distance plateau, level out, coverge towards what is around 15.6.
the amount it changes decreases with each step.

that is because the CEH is at 15.6. If we want to send a message today to a galaxy and have it get there NO MATTER HOW LONG IT TAKES then the galaxy can not be farther than 15.6 at this time. Have to go so must leave the post unedited but hope it's clear and someone will make the table and see the convergence beginning to happen. You can see that kind of thing in a table (with small steps) when you can't see it with a one-shot.

 

[Jorrie]

I have done an experimental version of A20 with the distance of the cosmic event horizon added. It required increasing the number of integration step significantly, because it has to calculate far into the future, so the calculator becomes slower.

It is not all that well tested, so I want to get reaction before finalizing and uploading it for direct access. Here is a screenshot for discussion. Due to the increased number of integration steps, some of the values are marginally different. Still evaluating for possible errors.

I think we must carefully consider the descriptions of the (live) info popups so as to not cause confusion with the generalized meaning of distance columns.
Edit: my proposal

D_now
"If positive: present proper distance of a source from which we now receive light with a wavelength stretch S. If negative: present proper distance of a target which will receive our present signals with a wavelength stretch 1/S. Proper distance is like measuring cosmic distances on a hypothetical 'freeze-frame' (no expansion) by means of radar, measuring rods, or similar."

D_then
"If positive: past proper distance (at emission) of a source from which we now receive light with a wavelength stretch S. If negative: future proper distance of a target which will receive our present signals with a wavelength stretch 1/S. See D_now info for definition of proper distance."

D_hor
"The cosmic event horizon, which is the largest distance (at time of emission) between an emitter and receiver that light can ever bridge. At larger distances, accelerating expansion prevents light from reaching the receiver."

I will appreciate suggestions for improvement of clarity.

 

[marcus]

It's great to have a Cosmic Event Horizon column! the CEH must surely be one of the most interesting dimensions of cosmology. I had assumed that the computational cost would be too great to include a calculation of it---that it would prohibitively slow the generation of the table. Is it really OK to include that much numerical integration? I guess it must be OK! Jorrie may have found a way to make the whole thing more efficient and thus practical.

Right now I can't think of suggestions about wording, maybe someone else can suggest some clear concise phrasing to put in the "info" popups. AFAICS they are already pretty clear and helpful.

 

[marcus]

My wife and I just got back from a short vacation at a perfect spot on the California coast. It's a pleasure to see this tentative new version of Jorrie's A20 calculator that actually calculates the CEH at any time in the past present future. And let's you vary the inputs parameters of the standard cosmic model and get different everything (including different CEHs).

I still haven't gotten any sense as to how practical it is to build in this addtional computation load. Or if it does slow the calculator down whether Jorrie will choose to pay that tradeoff cost in order to get the new feature. Right now it's pretty amazing. I just generated a table with over 100 rows and the output was instantaneous.

The table went from S=10 to S=0.01 in steps of ΔS = 0.09, so it had over 110 rows, more than I feel like printing in this post. But it went from the formation of the first galaxies (around year 560 million) out to year 88 billion: far in future when distances are 100 times what they are now. So a sweeping panorama of history---from remote past to distant future---and a lot to think about. And the calculator generated it instantly. If I had constructed such an online device would I be willing to add an extra function that would slow it down? I don't know. We will see what happens

In the meanwhile let's see if I can copy Jorrie's brief sample "screenshot" that he posted yesterday---with the extra CEH column.

There is a lot you can get out of even a short table like this. For example it shows us sending a message NOW which eventually arrives at a galaxy when the galaxy is 1550 billion lightyears from here OK? Now suppose you want to know how fast is the distance to that galaxy growing on the day that the message from us arrives?

How fast is that distance 1550 billion ly growing? That's easy with the A20 calculator's table because it gives the THEN Hubbletime. You can always divide any distance by the contemporaneous Hubbletime and it gives the distance's growth speed.

You can see (if the screenshot prints in my post, if not look back at Jorrie's post) that the THEN Hubbletime is 16.3 billion years.
So you just divide 1550/16.3 and it gives the recession speed as a multiple of speed of light on the day that our flash of light arrives at the designated galaxy to which it was addressed.
If you are new to the subject you might want someone to explain how it can possibly get there* and in that case since there are quite a few people around the forum who can explain that you simply need to ask. Or you may already have read the explanation in the Sci Am article by Lineweaver and Davis (the "charley" link at the foot of this post).

Another thing readers might like to imagine is how it will look to us far in future as galaxies sail over the Cosmic Event Horizon. A bit like watching things falling in thru the event horizon of a black hole. there are analogies between a black hole EH and the cosmic EH.

*i.e how could info we send at speed of light ever reach a galaxy which is receding at 95c.
We were within that galaxy's event horizon on the day we sent the message, as likewise it was within ours (see table) and so it CAN reach them eventually---that's what the CEH means. The challenge is to understand it well enough that you can imagine how this happens, and also how it fails to happen if the galaxy is just a little bit farther than CEH on the day we send signal.

As an afterthought just to be emphatically quantitative about it the above table shows that the TODAY value of the CEH is 15.622 and the today distance to the galaxy is 15.489. (So you are using 2 rows of the table to see that 15.489 < 15.622, and that is what makes it possible for our signal to reach them. That being what event horizon means. And it works both directions---they could right now be sending a flash of light in our direction, from the brink so to speak, like goodby from something close to falling forever into BH. A message destined to reach us but not for 88-14 = 74 billion years. We are within their CEH,as they within ours, but not for too much longer (again see table, as a rough estimate I'd guess two or three hundred million years should see them over the edge ;-)

reminder:https://www.physicsforums.com/showthread.php?p=4069097#post4069097
Update: today Jorrie put version A22 online:
https://www.physicsforums.com/showthread.php?p=4072363#post4072363
It turns out to be practical to include a column showing the cosmological event horizon!
A22 does quite a bit more numerical integration but the output, even with a fairly large table, is still instantaneous. the additional information provided by the A22 version is a valuable aid to seeing what's going on in the expansion process.

 

[marcus]

Year 88 billion from start of expansion is when the cosmic event horizon (now about 15.6 billion lightyears) get so close to the limit of 16.3 that it is AT the limit to within reasonable accuracy. That's when intergalactic distances will be 100 times what they are now.

And year 88 billion is about 74 billion years in the future. So that is what I should have said at the beginning. But perhaps it doesn't matter.

This table-maker online calculator keeps getting improved and the last time I posted version A22 had just come out, which is now already surpassed. We are now at version A25 and it has at least one very handy new feature. So I should give the link.
http://www.einsteins-theory-of-relativity-4engineers.com/CosmoLean_A25.html

The feature I'm thinking of is that rather than having to figure out an appropriate step size you can simply decide on how many rows you want, like 30 (a nice size table), and enter minus that for the step size. It will figure out what size step to use going from on row to the next so that (given your specified range) you will get the desired number of rows.

So say you pick the range S=10 to S=0.01, which means going from the era when the first little proto galaxies were forming and distances were 1/10 of today to the era when distances will be 100 times today's---in effect from around year 0.56 billion to year 88 billion.

And say you want 30 rows, so you type in
upper limit = 10
lower limit = 0.01
stepsize = -30
and just press calculate.

Here is a visual picture of what you will be getting in tabular form:

Looking over to the right is looking deep into the past to when distances were 1/10 today's.
For instance if we see a galaxy with S=9 meaning that its light comes to us wavestretched 9-fold you can see from the figure that the galaxy NOW is 30 billion lightyears away, as the green curve shows.
How far was it back THEN, when it was as we see it and emitted the light? Well, distances and wavelengths have enlarged by a factor of 9 since then. So it must have been 30/9=3.33 billion lightyears away back then. And that is what the purple curve shows.

Another interesting thing you can read off the figure, about the S=9 era, is that the cosmic event horizon was only 5 billion lightyears back then!
Any galaxy from that S=9 era which we are now able to see MUST have been closer than that, at the time.

That's looking over to the right side of the figure, high values of S means deeper into the past.
On the other hand, looking over to the left side we have a compressed picture of the future, from the present S=1 era out infinitely far in the future at S=0. You can see the black curve of time, the yearcount, blowing up to infinity. You can see the distances to any galaxy we can reach with a flash of light we send today. Its distance from us NOW (green) and its distance from us THEN (purple) when the flash arrives. You can see that distance at arrivaltime also going off to infinity, along the purple curve. And the distance from us now approaches a finite limit which is today's cosmic event horizon. Any galaxy beyond that limit we have no hope of sending light to, as of today. Though past light from us, or from our Milky Way galaxy, may already be reaching them or in the process of getting to them, assuming they were within event horizon range when the light was emitted. That event horizon distance is shown by the sky blue curve.
It's current value is over the S=1 mark representing the present. Going up the S=1 vertical line you can see it is slightly over 15 billion light years currently (I think the exact figure is about 15.6)
There's a lot more that you can read from the figure, or from the corresponding table.
Recall that to get the table you just type upper=10, lower=.01, and stepsize=-30, say, to specify the desired number of rows.

 

[marcus]

BTW the title should have said look 74 billion years into future, not 88 billion. The calculator goes out to year 88 billion of the expansion, but we are already nearly at year 14 billion. Careless error.
Mainly here I want to copy a couple of Jorrie's posts that provide technical background to the A25 calculator. That way we will have them in this thread for reference and aren't so likely to lose track:

===quote Jorrie post#4066605===
The latest version (as in Marcus's signature) is CosmoLean_A20, which adds an 'Introduction' button with some hints for usage. It is supposed to be fairly stable now and it is perhaps time to give an idea of the underlying formulas and conventions. It follows the development of the 13.9/16.3 simplified model proposed by Marcus, but with inclusion of the early stage radiation energy density.

The basic input parameters are:
present Hubble time $Y_{now}$, long term Hubble radius $Y_{inf}$ and the redshift for radiation/matter equality $z_{eq}$. Since the factor $z + 1$ occurs so often, an extra parameter $S = z + 1$ is defined. From these, the Friedman equation terms for the cosmological constant, radiation and matter can respectively be found for a perfectly flat LCDM model.

$$\Omega_\Lambda = \left(\frac{Y_{now}}{Y_{inf}}\right)^2, \ \ \, \Omega_r = \frac{1-\Omega_\Lambda}{1+S_{eq}}, \ \ \, \Omega_m = S_{eq}\Omega_r$$

The 'heart' of any simple cosmological calculator is the time variable Hubble constant $H$, which comes from the Friedman equation as:

$$H = H_0 \sqrt{\Omega_\Lambda + \Omega_r S^4 + \Omega_m S^3}$$

For perfect flatness, it can be expressed as

$$H = H_0 \sqrt{\Omega_\Lambda + \Omega_m S^3 (1+S/S_{eq})}$$

It can be interpreted in terms of the "13.9/16.3 factors" as follows: $\Omega_\Lambda = 0.7272$ and $\Omega_m (1+S/S_{eq})= 0.2728$, which of course sum to 1 (required for perfect flatness). It also shows at a glance how the influence of the various energy densities changes with S. Since S_eq ~ 3350, radiation dominated when S > 3350 and matter dominated for S < 3350, until such time as $\Omega_m (1+S/S_{eq}) < 0.7272$, when the cosmological constant started to dominate the equation.

From H, the following calculator outputs are readily available:

Hubble time $$Y(a) = 1/H$$

Cosmic time $$T(S) = \int_{S}^{\infty}{\frac{dS}{S H}}$$

Proper distances to a source at stretch S, "now" and "then" respectively,

$$D_{now} = \int_{1}^{S}{\frac{dS}{H}},\ \ \ \ D_{then} = \frac{D_{now}}{S}$$

The integration for T(S) to S_infinity is problematic, but is usually stopped at a suitably high S (effectively close enough to time zero).

In principle, the equations can be used for projecting into the future as well. This has been "secretly" sneaked into version A20. If you want to try it out, enter 1 into S_upper and 0.1 into both S_lower and S_Step. Note the time going to some 50 Gy, T_Hubble to around 16.3 Gy and the distances to negative values.

As Marcus has pointed out before, D_now for this scenario is the present distance to a target that will receive our signals with a wavelength stretch S at future time T(a). D_then means the proper distance of the target when they eventually receive our signal, obviously 1/S times farther.

This 'trial feature' can go down to S = 0.01 in steps of 0.01, but not lower at this time.
==endquote==

This link will get into the midst of this series posts:
https://www.physicsforums.com/showthread.php?p=4072363#post4072363
They were written during successive stages of development, e.g. A20, A22,..A25, and describe useful features as they were added. The series of posts extends beyond the two quoted here.

==quote Jorrie post#4072363==
Hand-in-hand with the 'future option' goes the cosmic event horizon. It has been included in CosmoLean_A22.

For completeness, I'll repeat the prior post's equations together with D_CEH.

Given present Hubble time $Y_{now}$, long term Hubble time $Y_{inf}$ and the redshift for radiation/matter equality $z_{eq}$. Since the factor $z + 1$ occurs so often, an extra parameter $S = z + 1 = 1/a$ is defined, making the equations neater.

$$\Omega_\Lambda = \left(\frac{Y_{now}}{Y_{inf}}\right)^2, \ \ \, \Omega_r = \frac{1-\Omega_\Lambda}{1+S_{eq}}, \ \ \, \Omega_m = S_{eq}\Omega_r$$

Hubble parameter
$$H = H_0 \sqrt{\Omega_\Lambda + \Omega_m S^3 (1+S/S_{eq})}$$

Hubble time, Cosmic time
$$Y = 1/H, \ \ \, T = \int_{S}^{\infty}{\frac{dS}{S H}}$$

Proper distance 'now', 'then' and cosmic event horizon
$$D_{now} = \int_{1}^{S}{\frac{dS}{H}}, \ \ \, D_{then} = \frac{D_{now}}{S}, \ \ \, D_{CEH} = \frac{1}{S} \int_{0}^{S}{\frac{dS}{H}}$$

This essentially means integration for S from zero to infinity, but practically it has been limited to $10^{-7} < S < 10^{7}$ with quasi-logarithmic step sizes, e.g. a small % increase between integration steps.
==enduote==

 

[marcus]

We are now up to version A27 of Jorrie's calculator. It has had a lot of improvements and maybe is nearing final version.

I want to use it to answer a question. It is now year 13.7 billion of the expansion and the stretch factor of the CMB is about 1090---the redshift is always 1 less than the stretch ratio so if you like redshifts just subtract 1.

What will the CMB stretch be in year 17 billion? What will the CMB stretch factor be in year 19 billion?

And what will the radius of the SOURCE SHELL--the spherical surface of last scattering--be at those future times. Now according to Jorrie's calculator the present radius is 45.9 Gly.
And when the light was emitted it was 42.1 Mly.

I don't have time to explain but using the calculator I found that in year 17 billion the stretch will be 1362 and in year 19 billion it will be 1557.

Now what is really interesting to me is the DISTANCE to the hot gas source matter back when the light was emitted. D_emit

Jorrie calculator says that distance for the PRESENT CMB source matter was 42.1 Mly.
Not very far only 42 million light years. What about in the FUTURE?

From the calculator I get that in year 17 billion D_emit will be 44.8 million ly.
That is, then we will be getting CMB light NOT as now from stuff that was 42.1 Mly from here, but from stuff that was 44.8 Mly.

And in year 19 billion I see that D_emit will be 46.1 Mly.

I'll go thru the arithmetic of this later, have to do something else. But anyone interested can click on
http://www.einsteins-theory-of-relativity-4engineers.com/CosmoLean_A27.html and check it out.

Put in 1 and .6 for the upper and lower limits of the stretch factor. Put in 0.1 for the step and you will see part of what I just said.

 

[Naty1]

Thank you for the chart and explanation in post #9.

That gets my vote for the best post of all time...

 

[marcus]

Thank you for the chart and explanation in post #9.

That gets my vote for the best post of all time...

I'm delighted you found it good! It's really thanks to Jorrie--I guess everybody realizes that by now. I don't know how he gets the calculator to generate charts like that, with different color curves. I have a vague feeling that he explained to me at one point how to get charts, but I didn't take it in (a limited capacity for new information )

The calculator is the "A27" link in the signature at the end of this post.

It's a really nice calculator.

I should explain the arithmetic used in my preceding post. The question was as follows.

Now (Year 13.7 billion) we get CMB stretched 1090-fold from matter that was 42 Mly away.
The question was what about in future, like in Year 17 billion or 19 billion?
The answers calculated with the Jorrie online device were:

In Year 17 billion we'll be getting CMB stretched 1362-fold from matter that was 44.8 Mly away.

In Year 19 billion we'll be getting CMB stretched 1557-fold from matter that was 46.1 Mly away.

The source shell for the CMB gradually inches out--the distance of that matter from our matter at the epoch of the brief flash has to be larger. Because we GOT the flash from today's source shell and tomorrow we will be getting the flash from a slightly more distant(on average) shell which of course took a day longer to get here.

So how were those numbers calculated?
===========================

first off you put upper limit = 1 and lower limit say .6, and step .1
that tells you that stretch = 1 corresponds to NOW i.e. Year 13.755 billion.
and stretch = .8 corresponds to Year 17 billion
while stretch = .7 corresonds to Year 19 billion.
Then you just say 1090/.8 = 1362
and 1090/.7 = 1557
Does that make sense? If anybody is reading who wants that explained please say.
===========================

And the other thing is to find the distance to the matter back then at time of emission. We can find Dnow and then divide by 1090. By Dnow I mean D_13.755.
Think of us as a waystation. The light from the flash has already traveled 45.9 when it passes us and to get to the people in Year 17 billion it still has to how far? In Dnow terms.
If you did that tabulation with step = .1 I suggested earlier, you know that the two distances we need are 2.9 and 4.4
those are the addition distances the light has to travel to get to the people in Years 17 and 19 billion. But it already traveled 45.9 by the time it got to us. (In Dnow terms.)

(45.9+2.9) billion/1090 = 44.8 million
(45.9+4.4) billion/1090 = 46.1 million

That's where the numbers in the earlier part of the post came from.
Again, this may not be a sufficient explanation so if puzzled by anything please ask.

I think the main thing is to click on the calculator and actually put in upper=1, lower=.6, step=.1 and look at the resulting tabulation and think about what the numbers you see in the table mean. that is, the stretch factors .8 and .7, and the times 17 and 19 billion, and the now-distances 2.9 and 4.4

Plus earlier if you were unfamiliar with the now distance 45.9 Gly to the flash source matter, you might have put in upper=1090, step = 0 (a quick way to get a table with only one row)

 

[Tanelorn]

Hi Marcus, thanks for this very interesting calculator.

I just wanted to ask about the Demit number is this the Dthen parameter in the calculator?

Also would you be able to set up a version of this with all the necessary parameters in place for the year 88 Billion? It is just that I am not sure which and how many parameters to change. I am just curious what distance the radiation first was that we will receive it then. It would be interesting to see a 20 point chart of how this distance changes from say year 0 to 88 Billlion.

Thanks again, great job Marcus.

 

[marcus]

Hi Marcus, thanks for this very interesting calculator.

I just wanted to ask about the Demit number is this the Dthen parameter in the calculator?

Also would you be able to set up a version of this with all the necessary parameters in place for the year 88 Billion? It is just that I am not sure which and how many parameters to change. I am just curious what distance the radiation first was that we will receive it then. It would be interesting to see a 20 point chart of how this distance changes from say year 0 to 88 Billlion.

Thanks again, great job Marcus.

It's great you like it! It's really an extraordinarily cool online device IMHO. Jorrie gets all the credit. Actually it's a credit to Physicsforums that the forum provides an environment for stuff like this to happen. Just to be clear, I personally had no hand in the building of the calculator. I'd like to encourage it's wider use.

The calculator is the "A27" link in my signature at the end of this post.

If you want a table that goes from NOW (stretch factor = 1) to Year 88 billion (stretch factor 100) and which has, say 21 rows,
then you just have to type in upper=1, lower=.01, and step = -20

There are popups that will appear to help interpret the numbers in the table. Jorrie put them in, as he did with everything.

I'll say in my own words Dnow is the distance between us and the other matter, today (if you could stop expansion to allow for it to be measured.)
If the Dnow number is positive, think of a signal from the other matter coming towards us
If the Dnow number is negative, think of the signal from us to the other stuff, imagine creatures there to receive it

Dthen is the distance between us and the other matter on the day when the light was emitted (if distance positive) or received (if distance negative).

If Dthen is positive that means the signal is coming towards us from the other matter, so "then" means when it was emitted by the other matter.
If Dthen is negative it means the signal is going away from us towards the other matter, so "then" means when the other "people" receive it.

Far in the future they will be be very far away when they finally receive our message, because of all that expansion while the signal is en route to them. You can see that if you just put in the suggested numbers upper=1 lower=.01 step=-20.
(or however many steps in the table you want, 10, 20, 30...whatever).

The HORIZON distance in the last column of the table is something I find fascinating. If someone checks out the other stuff and then gets to wondering about that column, it might be fun to discuss some.

 

[Tanelorn]

Thanks Marcus, Sorry I am still having trouble.
I was looking for a column which shows the distance then from us to where the CMBR light was emitted and how this distance changes with time from t=0 to t=88 Billion years. None of the columns seem to have the 41 Million light years distance in there.

 

[Jorrie]

If you did that tabulation with step = .1 I suggested earlier, you know that the two distances we need are 2.9 and 4.4
those are the addition distances the light has to travel to get to the people in Years 17 and 19 billion. But it already traveled 45.9 by the time it got to us. (In Dnow terms.)

(45.9+2.9) billion/1090 = 44.8 million
(45.9+4.4) billion/1090 = 46.1 million

That's where the numbers in the earlier part of the post came from.
Again, this may not be a sufficient explanation so if puzzled by anything please ask.

Marcus, the latter two values are correct as defined for the calculator, but they may need a little more explanation. When we speak about distances "now", we measure with our local yardstick, which is essentially the local speed of light multiplied by the time it would take light to traverse the distance, provided that we could 'freeze' the expansion of space now (as you have said).

When we speak about distances "then" we use the same yardstick (our present one), but we 'freeze' the expansion at the time of emission. For the past, that was obviously "then" and not "now", because we are not the emitting party. For the future, we are the emitting party, so we freeze the expansion "now" and and D_now is how far light has to travel to reach the other civilization in a static cosmos.

Tanelorn's question may arise from the fact that for the future D_then, we have to freeze the expansion when the light reaches the other civilization and find out how long the light had to travel in an expanding cosmos. So, it is important to note that one cannot always equate D_emit with D_then. This is a price we pay for having the past and the future together in one table.

 

[marcus]

Marcus, the latter two values are correct as defined for the calculator, but they may need a little more explanation...

I am just so pleased with this calculator! It's versatile like a swiss jackknife, you can do a lot of things with.

And that leads to opportunities to explain more stuff, like what you just did. For instance the idea of proper distance (distance at a certain moment with expansion stopped).

Also I guess I should be restrained and not try to bring in TOO MUCH. Introduce concepts slowly. But it's tempting to also mention the idea of the COMOVING DISTANCE which is simply Dnow. The comov. distance of some matter is a way of tagging the matter with a serial number that never changes. So it can be useful. By convention it is the now distance (as of today).

When I did those simple calculations before, conceptually I was working in comoving distance, and then at the end, converting to proper distance at "recombination time" simply by dividing by 1090. That is always how one converts comoving distance to distance at the moment the CMB flash was emitted.

==quote==
... suggested earlier, you know that the two distances we need are 2.9 and 4.4
those are the addition distances the light has to travel to get to the people in Years 17 and 19 billion who are going to receive it. But it already traveled 45.9 by the time it got to us. (In Dnow terms.)

(45.9+2.9) billion/1090 = 44.8 million
(45.9+4.4) billion/1090 = 46.1 million
==endquote==

I guess I am actually talking to people who might be reading (not to you Jorrie). You have to click on the "A27" link and put in something like
upper=1
lower=.6
step=.1
and actually see that for those Years the Dnow distances 2.9 and 4.4

So after the light has covered a comov. distance of 45.9, and has passed the way station (us) and is racing on towards the people who are to receive it in Year 17 billion,
then how far, in comov. terms, does it have to travel?
Well it is just as if we sent them a message today--a flash of a different color to travel neck and neck all the way to them. And the comov. distance from us to them is 2.9

So to get the total comov. diet you just add 45.9 and 2.9. That is the comov. dist between them and the source matter of the CMB which they will see in Year 17 billion.

Now since it is the comov. distance you just have to divide by 1090, and bingo.

Jorrie forgive me if I'm revealing too much enthusiasm, but AFAIK this is unique. I don't know of any online cosmo calculator that has future as well as past, and none of the others give the Event Horizon. In fact they don't do half the things this one does. There's only one other I've seen that tabulates. (A recent one by someone at Oxford in the UK, as I recall.)
The trouble is we have a distribution bottleneck. More beginning astronomy students should get to use it. I don't know how to get the word out. Maybe there are some active academics here at PF who would be willing to pass the link along to colleagues in the astronomy department.

 

[Naty1]

None of the columns seem to have the 41 Million light years distance in there.

edit: oops, did not realize we have a page two...and two more posts befoe this. [We shall have to slow Marcus down as in "Curb that enthusiasm." as he is posting faster than I am reading [LOL]...

If I understand what I am looking at in the chart, 41/42 mly is way,way off to the right, via the purple curve, 'distance then'..

As I recall that is about 380,000 years after the big bang and a redshift of about 1090.

[The chart uses S =1+z, and only goes to S =10, redshift of z = 9. ]

In fact I started to try to calculate the distance at z =10, and never found a simple formula...I suspect that's why we have the calculator...

 

[marcus]

edit: oops, did not realize we have a page two...and two more posts befoe this. [We shall have to slow Marcus down as in "Curb that enthusiasm." as he is posting faster than I am reading [LOL]...

If I understand what I am looking at in the chart, 41/42 mly is way,way off to the right, via the purple curve, 'distance then'..

As I recall that is about 380,000 years after the big bang and a redshift of about 1090.

[The chart uses S =1+z, and only goes to S =10, redshift of z = 9. ]

In fact I started to try to calculate the distance at z =10, and never found a simple formula...I suspect that's why we have the calculator...

It's great you are using the calculator, I find it really "empowering" (as they say).

As you know, and most other readers as well, YOU decide the range that the table covers. for example if you want it to cover all of history since the CMB flash (which gets stretch 1090) and if you want the table to have 10 downsteps from there to the present (i.e. 11 rows) then you put in:

upper=1090
lower=1
step= -10

You can also make it show more, or fewer, decimal places in the answers. So it can give more precision in some columns and round off in other columns. You just type the number of decimal places you want the answer to have, in that column, in the box at the head of the column.

 

[marcus]

Here's a result I just now got, to demo the feature where you control the precision. I set it to show 1 decimal place for the Stretch, 4 for the scalefactor a, 6 for the Year, 4 for the Hubbletime, 1 for Dnow, 4 for Dthen, and 3 decimal places for Dhor the event horizon distance:

S	a	T	       T_Hub	D_now	D_then	D_hor
1090.0	0.0009	0.000381	0.0006	45.9	0.0421	0.056
981.1	0.0010	0.000455	0.0008	45.8	0.0467	0.063
872.2	0.0011	0.000554	0.0009	45.7	0.0524	0.070
763.3	0.0013	0.000691	0.0011	45.6	0.0598	0.080
654.4	0.0015	0.000889	0.0015	45.5	0.0695	0.093
545.5	0.0018	0.001196	0.0019	45.3	0.0830	0.112
436.6	0.0023	0.001712	0.0027	45.0	0.1032	0.139
327.7	0.0031	0.002704	0.0043	44.7	0.1363	0.184
218.8	0.0046	0.005103	0.0080	44.0	0.2012	0.273
109.9	0.0091	0.014809	0.0227	42.6	0.3875	0.530
1.0	1.0000	13.754712	13.8999	0.0	0.0000	15.622

I checked the calculator's "copy/paste friendly version" box to get a table output in a format that I could copy and paste easily. But the normal output looks better, in the grid designed for it. So try it yourself
just set upper=1090, lower=1, step=-10
1090 being the stretch factor for the CMB flash and 1 being the stretch factor for the present (i.e. no change in distances or wavelengths, the identity)

 

[Jorrie]

The trouble is we have a distribution bottleneck. More beginning astronomy students should get to use it. I don't know how to get the word out. Maybe there are some active academics here at PF who would be willing to pass the link along to colleagues in the astronomy department.

I've got some feedback that our use of the Hubble times Ynow and Yinf as the primary input parameters are off-putting to some teachers. They say it brings in the idea of the Hubble time too early for beginners and they would like to start with the Omegas; students do not find the Y's in published results, only the Omegas. Some also prefer z rather than S as an input, but this is not really an important issue. Maybe having negative z for the future is more intuitive than our 0 < S < 1?

Actually in my calculator I immediately convert the Ys to Omegas, so apart from the user interface and "front-end", nothing else need to change. I will look into the possibility of a B-model, returning it to the legacy input parameters, giving people a choice.

What do you think?

 

[Jorrie]

In fact I started to try to calculate the distance at z =10, and never found a simple formula...I suspect that's why we have the calculator...

Yes, if you look at the equations in https://www.physicsforums.com/showpost.php?p=4072363&postcount=29, they all have integrals - hence the need for a numerical integrator...

(Now CosmoLean_A27, no longer A22).

 

[mfb]

Some different scaling for the scale factor entries would be interesting - maybe equidistant logarithm or equidistant scale factor (+optional?). If you want to look at the range of 1090 to 1, for example, you just have a single entry in the stelliferous era, unless you want 100+ lines as output. Entries like (1090, 545, ..., 2, 1) +- rounding errors would be more relevant I think.

 

[Jorrie]

Some different scaling for the scale factor entries would be interesting - maybe equidistant logarithm or equidistant scale factor (+optional?). If you want to look at the range of 1090 to 1, for example, you just have a single entry in the stelliferous era, unless you want 100+ lines as output. Entries like (1090, 545, ..., 2, 1) +- rounding errors would be more relevant I think.

Yes, good idea. I will look into that as an option during the next update.

What do you think of working with Hubble time inputs vs. standards density parameters and Hubble constant as input?

 

[mfb]

I think if you want to change those parameters, you usually know how to do that. Density parameters are probably easier to manipulate, however.

 

[Jorrie]

Some different scaling for the scale factor entries would be interesting - maybe equidistant logarithm or equidistant scale factor (+optional?). If you want to look at the range of 1090 to 1, for example, you just have a single entry in the stelliferous era, unless you want 100+ lines as output. Entries like (1090, 545, ..., 2, 1) +- rounding errors would be more relevant I think.

I have done a first attempt at an optional nonlinear scaling of the stretch (scale) factor. It divides S by any S_step larger than 1, with 2 probably the most useful. There is a tick box in the latest version, which I have renamed to TabCosmoC1 (for Tabular Cosmological Calculator).

After a bit of forum testing and perhaps tuning, we can make it more visible again.

 

[Naty1]

Jorrie...this model: cool. Thanks for your efforts.!

 

[mfb]

Very nice, thanks.

Yinf standard is 16.3, but the tooltip gives 16.9 as best fit?

 

[marcus]

Indeed it is very nice! Here's something I just tried with the nonlinear step box checked.
I wanted 20 steps down from 1090 to 1 (from recombination to present)
so I put this into google calculator:
"1090^.05"
and got 1.41863714

That is the 20th root of 1090.
So I put in S upper = 1090
S lower = anything less than 1 would do, I happened to put 0.15
step = 1.41863714 (just pasted in from the calculator)

S	a	T	T_Hub	D_now	D_then	D_hor
1090.00	0.001	0.000	0.001	45.890	0.042	0.056
768.34	0.001	0.001	0.001	45.617	0.059	0.080
541.61	0.002	0.001	0.002	45.280	0.084	0.112
381.78	0.003	0.002	0.003	44.870	0.118	0.158
269.12	0.004	0.004	0.006	44.373	0.165	0.223
189.70	0.005	0.006	0.010	43.773	0.231	0.313
133.72	0.007	0.011	0.017	43.051	0.322	0.439
94.26	0.011	0.019	0.029	42.186	0.448	0.613
66.44	0.015	0.032	0.049	41.151	0.619	0.855
46.84	0.021	0.054	0.082	39.913	0.852	1.186
33.02	0.030	0.092	0.140	38.436	1.164	1.638
23.27	0.043	0.157	0.236	36.673	1.576	2.248
16.40	0.061	0.265	0.399	34.571	2.107	3.061
11.56	0.086	0.449	0.675	32.066	2.773	4.125
8.15	0.123	0.759	1.140	29.084	3.568	5.486
5.75	0.174	1.282	1.917	25.541	4.445	7.167
4.05	0.247	2.159	3.200	21.353	5.272	9.133
2.86	0.350	3.607	5.224	16.474	5.770	11.247
2.01	0.497	5.913	8.091	10.988	5.460	13.230
1.42	0.705	9.312	11.326	5.272	3.716	14.739
1.00	1.000	13.755	13.900	0.000	0.000	15.622
0.70	1.419	18.895	15.324	-4.341	-6.158	16.025
0.50	2.013	24.378	15.937	-7.598	-15.292	16.178
0.35	2.855	29.998	16.170	-9.954	-28.419	16.225
0.25	4.050	35.670	16.254	-11.630	-47.105	16.254
0.17	5.746	41.360	16.284	-12.816	-73.637	16.284

 

[Jorrie]

Yinf standard is 16.3, but the tooltip gives 16.9 as best fit?

Yes, tooltip in error. Thanks for heads-up, will fix.

 

[Jorrie]

I wanted 20 steps down from 1090 to 1 (from recombination to present)
so I put this into google calculator:
"1090^.05"
and got 1.41863714

I thought about putting something like this in the calculator, but in order to retain flexibility, it clutters the relatively 'clean' look and feel a little. Maybe I should put it in the accompanying tooltip...

Edit: Done. Also fixed the tooltip mistake.

If your cache prevents modified version from loading, try TabCosmoC2.html. It is the same as C1.

 

[marcus]

One of the luxuries associated with using the new online Cosmic Tabulator is you can (in seconds) get a whole history of the universe from recombination (the origin of the Background) down to today and on far into the future.
I just tried this to see how it would look if I set it to use 30 steps to get from recombination to Now, and then let it do a dozen or so steps beyond that into the future.

What I like about this (besides that it is very quick to get once you decide you want to start at S=1090 and get to the present in 30 steps) is that you see in context a lot of the numbers that we are always hearing about--that keep coming up in Cosmo Forum threads. For instance right in the first row you see that the CMB was emitted in year 380,000, by matter that was then 42 million LY away, and is now 45.89 billion LY from us.

And you see, from the bottom row, that both the distance to the cosmic event horizon AND the Hubble distance are converging together to 16.3 billion LY. In fact that's what they'll essentially be by year 62 billion.

And the Hubble times give you a convenient handle on the expansion rates now, and in the past, and in the future.
You can see the present Hubble time (in the S=1 row) is 13.9 billion years---this means distances are increasing by 1/139 of a percent every million years.

Far in the future, when the Hubble time is stabilizing at 16.3 billion years, distances will of course be increasing by 1/163 of a percent every million years. Then again, if you check out the row around S=2.5 you'll see that, back around year 4 billion, expansion was considerably faster---around 1/60 of a percent per million years.

And the nice thing is you get to see how all these numbers gradually change over time.

S	a	T	T_Hub	D_now	D_then	D_hor
1090.0	0.001	0.00038	0.001	45.890	0.042	0.056
863.33	0.001	0.00056	0.001	45.714	0.053	0.071
683.80	0.001	0.00083	0.001	45.512	0.067	0.089
541.60	0.002	0.00121	0.002	45.280	0.084	0.112
428.97	0.002	0.00176	0.003	45.016	0.105	0.141
339.77	0.003	0.00255	0.004	44.715	0.132	0.178
269.11	0.004	0.00369	0.006	44.373	0.165	0.223
213.15	0.005	0.00532	0.008	43.985	0.206	0.280
168.82	0.006	0.00764	0.012	43.547	0.258	0.351
133.72	0.007	0.01095	0.017	43.051	0.322	0.439
105.91	0.009	0.01567	0.024	42.492	0.401	0.549
83.889	0.012	0.02239	0.034	41.861	0.499	0.685
66.444	0.015	0.03196	0.049	41.151	0.619	0.855
52.627	0.019	0.04555	0.069	40.350	0.767	1.064
41.683	0.024	0.06488	0.098	39.449	0.946	1.322
33.015	0.030	0.09233	0.140	38.436	1.164	1.638
26.150	0.038	0.13132	0.198	37.295	1.426	2.024
20.712	0.048	0.18667	0.281	36.013	1.739	2.494
16.405	0.061	0.26525	0.399	34.571	2.107	3.061
12.993	0.077	0.37676	0.567	32.950	2.536	3.739
10.291	0.097	0.53495	0.804	31.129	3.025	4.544
8.151	0.123	0.75926	1.140	29.084	3.568	5.486
6.456	0.155	1.07696	1.613	26.790	4.149	6.571
5.114	0.196	1.52614	2.277	24.220	4.736	7.794
4.050	0.247	2.15887	3.200	21.353	5.272	9.133
3.208	0.312	3.04392	4.454	18.176	5.666	10.540
2.541	0.394	4.26571	6.093	14.701	5.786	11.940
2.013	0.497	5.91316	8.091	10.988	5.460	13.230
1.594	0.627	8.05254	10.269	7.165	4.495	14.305
1.263	0.792	10.6898	12.304	3.428	2.715	15.101
1.000	1.000	13.7547	13.900	0.000	0.000	15.622
0.792	1.263	17.1291	14.965	-3.021	-3.814	15.929
0.627	1.594	20.6981	15.594	-5.540	-8.831	16.095
0.497	2.013	24.3777	15.937	-7.598	-15.292	16.178
0.394	2.541	28.1165	16.117	-9.255	-23.516	16.216
0.312	3.208	31.8857	16.208	-10.578	-33.934	16.230
0.247	4.050	35.6702	16.254	-11.630	-47.105	16.254
0.196	5.114	39.4625	16.277	-12.465	-63.743	16.277
0.155	6.456	43.2586	16.289	-13.127	-84.754	16.289
0.123	8.151	47.0568	16.294	-13.652	-111.28	16.294
0.097	10.291	50.8558	16.297	-14.068	-144.77	16.297
0.077	12.993	54.6553	16.299	-14.397	-187.06	16.299
0.061	16.405	58.4551	16.299	-14.658	-240.46	16.299
0.048	20.712	62.2551	16.300	-14.865	-307.87	16.300

Time now (at S=1) or present age in billion years:	13.7547

Again looking at the bottom row, to give an example of telling yourself stories to interpret the numbers, suppose today we decide to send a message (an intense flash of light) to a galaxy which is NOW 14.865 billion LY from us. there are lots of galaxies like that, most of those we see with a telescope are that far or farther. So imagine we pick out an especially pretty one and decide to send them a message today. How long will the message take to reach them? Well today is year 13.75 billion and the table says the message gets there in year 62.25 billion, so you do the arithmetic.
And how far will they be from us when the message actually "catches up" to them and gets there? The table says they will be 20.7 times farther than they are today, and that means they will be at a distance of 307.87 billion LY.

I keep the link to the calculator in my signature to have it handy. But here it is, a little more visibly:
http://www.einsteins-theory-of-relativity-4engineers.com/TabCosmoC1.html
As always, thanks Jorrie. It's neat.

 

[marcus]

I mentioned that if you look down to the last row of the table, around year 60 billion when distances are 20 times what they are at present you will see the important distance 16.3 billion LY.

Both the Hubble distance and the Cosmic Event Horizon distance are converging to that important 16.3---you can see that in the table very clearly.

That 16.3 billion LY distance is a physical substantive visible form of the cosmological constant Λ. Just take the reciprocal of that distance, and square it, and you have Λ.

There is nothing "dark" or mysterious about it. It was a mistake, I think, to ever refer to it as "dark energy". Technically it is a small intrinsic curvature which naturally appears in the Einstein GR equation and whose value was not known until recently (the 1990s). The value might have turned out to be exactly zero, but there was no reason it had to be zero, and as it turned out, it wasn't.

 

[Naty1]

What is this some sort of physicsforums 'quiz'...??
when I look at the bottom row of the table to follow your explanations I see these numbers...
I don't see all the numbers quoted in your text:

S a T T _Hub D_now D_then D_hor

1.594 0.627 8.05254 10.269 7.165 4.495 14.305

Am I getting THAT old??

edit: darn! I just realized there is a scroll bar along the right hand side of the table... problem solved!