Cosmo calculators with tabular output

[marcus]

I think I know now what the vertical dashed line labeled z=1.67 is supposed to be. With your numbers 14.0, 16.7, 3280, we get S=2.61 for the intersection of lightcone with Hubble radius.
That is, a galaxy we are observing today which was receding at c in the past when it emitted the light.

THAT is a galaxy which was subsequently inside the Hubble sphere, and then later was again outside.$${\begin{array}{|c|c|c|c|c|c|c|}\hline R_{now} (Gly) & R_{∞} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline14&16.7&3280&69.86&0.703&0.297\\ \hline\end{array}}$$ $${\begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&a=1/S&T (Gy)&R (Gly)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)\\ \hline2.6104&0.383083&3.970&5.7192&14.929&5.7192&11.828&11.737\\ \hline\end{array}}$$

So the vertical line for that galaxy does slice off a bit of the side bulge of the Hubble radius curve, just the way it appears in the figure. First it is outside Hubble sphere, then the sphere expands more rapidly than the galaxy is receding, and takes it in (for a while). Then its recession begins to dominate and it exits.

But that galaxy is not NOW at the Hubble radius. Your calculator says that its current distance is 14.929 Gly, not 14.0 Gly.

So instead of being labeled "z=1.67" the vertical dashed line probably wants to be labeled "z=1.61"
or S=2.61, and to be moved slightly over to the right so that it passes exactly thru the intersection of lightcone with Hubble radius. It will still slice off some of the bulge, on its way up, though slightly less of it.

OOPS! EDIT EDIT EDIT!
I see you relabeled that to say z=1.45. Now it makes sense, talking about a galaxy which is at comoving distance (now distance) Rnow = 14.0 Gly.

So multiplying that by the scale factor a(t) we get the past distance history of that galaxy
D(t) = Rnow a(t)

OK so that is a sample proper distance history. And you are going to take the slope of that.
And the slope should decline at first and then start increasing---the distance growth curve should have an inflection point where the slope is at a minimum. Which, as I recall, it does.

Yes! I checked on your table. S=1.636 is where the table minimum of the slope comes. Which is around year 7.6 billion. So that looks quite good. So I can see a real pedagogical benefit.

This is making a lot of sense now. I still don't have a definite opinion whether the 9th column pedagogical benefits outweigh the cost of having a more elaborate table. Probably it depends on who one expects to be the user.

 

[Jorrie]

I think I know now what the vertical dashed line labeled z=1.67 is supposed to be. With your numbers 14.0, 16.7, 3280, we get S=2.61 for the intersection of lightcone with Hubble radius.
That is, a galaxy we are observing today which was receding at c in the past when it emitted the light.

THAT is a galaxy which was subsequently inside the Hubble sphere, and then later was again outside.

So the vertical line for that galaxy does slice off a bit of the side bulge of the Hubble radius curve. First it is outside Hubble sphere, then the sphere expands more rapidly than the galaxy is receding, and takes it in (for a while). Then its recession begins to dominate and it exits.

But that galaxy is not NOW at the Hubble radius. Your calculator says that its current distance is 14.929 Gly, not 14.0 Gly.

I'll have to think about this a little more. A dotted vertical line represents a constant co-moving distance and, I think, a constant redshift over time. Galaxies below z ~ 1.67 must have entered the Hubble radius of the time and later exited it again. Now if the recession speed "then" must have been c when the galaxy entered the Hubble distance and again when it leaves it, there must be a single redshift that satisfies this condition for such galaxies. I could not find such a solution through the calculator, so now I'm a little confused.

What am I missing?

 

[marcus]

this is not a criticism of your Rnow a(t) column based on the vertical line labeled z = 1.45.
that made sense to me (and I edited my post) as soon as I saw you had relabeled it z = 1.45.

However there is a general comment to make. I think we need a notation for the maximum Dthen.

Dthen is the outline of the light cone, the galaxies we can be getting light from today. We've talked about its teardrop shape and its maximum girth, before.

If I call that maximum value of Dthen by the name "D_max", D_max = 5.7 or 5.8 depending on the parameters.
And the corresponding S_max = 2.61
And 2.61 x 5.7 = 14.9 billion light years, which is the comoving or now distance of a galaxy which emitted the light at the instant when it was receding at speed c, so that the light "stood still" at first, for a while and did not make any headway. this is a unique distance.

14.9 Gly is the unique comoving distance with that property.
====================

There's a slight possibility of confusion associated with plotting R_oda/dT in that it tracks the distance to something that is NOT ON THE LIGHTCONE.
Always in the past when we pick some S like S = 2.45 we are talking about a galaxy which we are getting light from today stretched by factor 2.45, and the distances in that row of the table tell us about the distance to that galaxy. So it's breaking with that precedent (for better or worse.)

You see the intersection of the horizontal line year 3.1 billion and the vertical S=2.45 is not on the red light cone curve. So we aren't getting any light from that galaxy that it emitted in years 3.1 and we aren't getting any light from it redshifted z=1.45. So the story with that galaxy is not LIKE the other stories we may be telling ourselves, habitually, about rows of the table. there is an "anomaly" in how we have to think about it.

But if you get back on the light cone, by using S=2.61, then your 9th column will be slightly different. The slope will start off at 1, at 2.61, and then it will decline as expansion slows, and then it will inflect and start increasing, and then it will reach 1 slightly BEFORE the present day, and then it will already be faster than light at the present. It will be greater than 1 at the present day. Which might not be a bad thing to show.
And you will be following the increasing distance of a real galaxy which we can see today, that is on our light cone. Because you start the vertical dash line at the INTERSECTION of the Hubble radius with the light cone.

I think that is pedagogically better, except that we have no NAME for SmaxDmax the comoving distance of the galaxy. Have to go, back later.

Back now. I guess one could fantasize teaching with this concept included in the kitbag. Explain that the past lightcone is onion-shape and the maximum proper radius we are going to call Dmax.
And then say that the COMOVING radius of the light cone (at its fattest) is going to be called Rcone. And we going to plot the recession speed history of a galaxy at R_cone.

this is a galaxy which, when it emitted the light we are getting, was RECEDING AT c!
So the speed number is going to be 1.
And that will be at S=2.61 and at a certain time, when it emitted the light, and when distance to it was increasing at c. So we picture that.
Smax x Dmax = 2.61 x 5.7 = Rcone = 14.9 Gly.
The thing which sent us photons that at first stood still is now 14.9 Gly from here, and we are getting the photons today.

And the 9th column record of R_coneda/dT starts at 1, when it emitted the light and was receding at c, and then it sags down because the thing's recession was slowing, and then it bottoms out and starts rising, and then it GETS TO ONE again, but it isn't the present yet. And by the time we come to present day it is actually receding a little bit faster than c. Good! That seems to work pedagogically.

However the cost is that one has to introduce a new concept R_cone the comoving radius of the past lightcone at its widest girth. So one has to weigh the cost. I'm interested enough I would like to see the 9th column used that way to get a sense of what it looks like.

I realize I haven't thought enough about this. The idea may have obvious flaws that I will only see later. But the 9th column (in units of the speed of light) does seem like an interesting idea. In future it would presumably show high multiples of the speed of light. And in past, before S=2.61.

 

[marcus]

I don't know why it's taking me such a long time to catch on. We were discussing Jorrie's idea of a 9th column that takes some distance as an example and watches it expand, during some interval of time. Column 9 would log the speed of the receding galaxy (as a multiple of c) and for earlier part of history this is slowing down while for later it is increasing. So we'd get to see this.

All cosmological distances (between CMB stationary pairs of observers) grow proportionally to a(t), which is a dimensionless number normalized to a(now) = 1.

If you multiply da/dT by the present-day Hubble radius, you get the recession speed history of a galaxy which is located at comoving distance R_now, and in the units we are using the speed comes out = 1. So the speeds are being expressed as multiples of c, i.e. in units of the speed of light.

What I'm undecided about (and periodically confused about) is whether one should allow optional flexibility about what one multiplies by. If you multiplied by HALF the Hubble radius instead, the speed numbers would come out half as big. And it would be a history of a galaxy only half as far away. So that seems consistent. Or you might multiply da/dT by 4/3 the Hubble radius and the speed numbers would be different accordingly, but they would be correct for a galaxy that is now 4/3 as far away.

The speed is always going to be expressed as a multiple of c, because of the units being used. Gy for time and Gly for distance. Maybe there should be a box where you put in a number like .5 or 1.333 and it says "da/dT will be multiplied by [box] times R_o the current Hubble radius, to give the recession speeds shown in column 9". And a tooltip says the speeds are given in units of c.

Still undecided about the desirable degree of flexibility.

 

[Jorrie]

A Wiki for Tabular Cosmo calculator user manual

With a complete overhaul of TabCosmoX taking shape and a draft user manual already posted by Mordred (to be updated for new 'release'), I was looking for a suitable Wiki-hosting site. Wikidot.com seems to be a good option for the manual. It allows collaboration with some control options and sports very good features, including Latex.

What do you think?

PS: WikiDot (or alike) also seems to be a good place for the calculator to be hosted, getting it off my private website, to where it may have more longevity...

 

[Jorrie]

New Look Tabular Calculator (LightCone)

The "complete overhaul of TabCosmoX" is completed and the new link is in my signature. It is now named "LightCone", proposed by Marcus. A sample screenshot is attached.

The main differences from TabCosmoX are the flexibility of selectable columns and a choice of default data sets (only WMAP and Planck at this time). More can be inserted if useful.

The main change is the column selector:

More columns can be added to the selection list with relative ease now.

Please report any usage issue or bugs detected.

 

[Mordred]

The "complete overhaul of TabCosmoX" is completed and the new link is in my signature. It is now named "LightCone", proposed by Marcus. A sample screenshot is attached.

The main differences from TabCosmoX are the flexibility of selectable columns and a choice of default data sets (only WMAP and Planck at this time). More can be inserted if useful.

The main change is the column selector:

More columns can be added to the selection list with relative ease now.

Please report any usage issue or bugs detected.

If you using IE 8, and see all the boxes in the column selector overrun each other instead of the view above. Check and make sure you have compatibility view turned off. Some IE 8 browsers may experience " a script is causing your browser to run slower than normal" error state no each time it asks to turn off script.
The script error appears to only occur on IE 8 and not other browsers. Jorrie is working on this issue.