Steps on the way to Lightcone cosmological calculator

[marcus]

Naty, I'm glad you saw the scroll bar and worked that out.

Just in case you are curious. the 73% we are always seeing is not a CONSTANT. It is a temporary figure that gives a handle on Lambda that depends on present conditions. Here is how to get it from the Hubble times 13.9 and 16.3 billion years.

Just calculate the ratio of their squares: 13.9²/16.3² = .73

13.9 billion years is information about the present, it is 1/H_o the reciprocal of the current fractional rate of distance expansion.

16.3 billion years is the corresponding thing way out in the future. THAT is the cosmological constant, in essence.
===========================

People who talk about Lambda as if it were a curvature that arises from a certain dark "energy" do not normally tell you what actual energy DENSITY it corresponds to in real terms like nanojoules per cubic meter. Normally they only tell you the temporarily valid handle 73%.

But I'll tell you how to get your hands on that energy density, in metric, just from the 16.3 figure.

[Footnote: If you know metric units you know that nanojoules per cubic meter is the same unit as nanopascal which is easier to say. N/m² =Nm/m³ = J/m³ because a Joule is a Newton-meter of work.]

Take the reciprocal of 16.3, square it, and multiply by 161 nanopascals.

That gives you the energy density people think corresponds to the cosmological constant.

"16.3^-2 * 161 nanopascal"

To use the google calculator, paste 16.3^-2 * 161 nanopascal into the window.
It will do the arithmetic and tell you 6.06 x 10^-10 pascals.
Which is 0.606 nanopascal.

That, precisely, is the constant energy density which people who imagine there is "dark energy" must think fills all of space. About 3/5 of a nanojoule per cubic meter.

But so far, all we have evidence for is a constant curvature term Lambda (the square of a reciprocal length) that appears in the Einstein equation, and improves the fit.

 

[marcus]

There is a fine new version of TabCosmo:
http://www.einsteins-theory-of-relativity-4engineers.com/TabCosmo5.html

What can I say? It is better than anything else on the internet for giving you a panoramic view of the evolution of the universe.

You can get a clear quantitative look at the expansion process from early times down to present and then out into the distant future. None of the other cosmo calculators I know of do this. so far. I think before too long we will see IMITATORS. There will be other online tabular cosmic calculators. It is just too good an idea.

Be sure you check the box where it says "S=1(exactly)". then your table will include the exact present.
then the number, like 10 or 20 or 30, that you put in the Step box will determine the number of steps from early universe (S=1089, the origin of CMB ancient light) down to the present (S=1).

The "S=1(exactly)" box is immediately above the top row of the table, right about the column labeled "D_then".
My favorite number of steps to put in is actually 29 ... I'll explain why later. But even with just 10 steps you get an interesting and enlightening table, and 20 is even better.
At each step, distances increase by the same ratio. Like for instance at each step they might go up by 20%, i.e. the get multiplied by 1.20. You can see that happening in the second column, the "a" or scale factor column. But in our case, the ratio is adjusted so that you land exactly on the present after the stated number of steps.

 

[Jorrie]

At each step, distances increase by the same ratio. Like for instance at each step they might go up by 20%, i.e. the get multiplied by 1.20. You can see that happening in the second column, the "a" or scale factor column. But in our case, the ratio is adjusted so that you land exactly on the present after the stated number of steps.

I guess one should say that the scale factor (not distance) increases by a constant ratio...

In any case, it is interesting to plot a graph on a log-linear scale to view all (or most of) the data in the table. Attached is an example, using data from the WMAP9 report, maximum likelihood, Table 2 of http://arxiv.org/pdf/1212.5226v2.pdf. It is done from calculator data copied to an Excel spreadsheet.

 

[marcus]

It's nice to see the new cosmological parameters being used. they are nearly the same but still recognizable as the latest set.

Handsome graph. I see you truncate the future at S=0.1 (when distances are 10 time what they are today.) That makes a visually nice graph and the range is something the mind can easily take in---just my subjective reaction. I like not making the future too vast in extent.

I like that the table hits S=1 exactly (when that box is checked) but on reflection don't care so much about it hitting a specific future S exactly, as well. Some users might though.

 

[marcus]

In another thread Jorrie posted a nice plot using the new cosmological parameters

For anyone who missed the discussion in Marcus' "88 billion year" sticky above, here is the latest version of the "inhouse" tabular cosmo-calculator, as also shown in Marcus' signature (TabCosmo5.html).

The main changes since September last year are: an easy method to get a logarithmic spread of redshifts (actually stretch S = z+1) and that the latest (2013) WMAP9 (combined) maximum likelihood parameters are now used. Please read the info tool-tips of the calculator for clarification of usage.

Here is a sample plot of data generated by the calculator, as copied into a spreadsheet.

Of particular interest from the visuals are the following observations:

1. The max value of D_then ~ 5.8, where D_then crosses T_Hub at S ~ 2.62. You will need 29 S-steps to spot this max precisely on a generated table.
   
2. The correspondence of T_Hubble and D_hor when S < ~0.3, where the cosmological constant completely dominates.
   
3. The 'straight' T-curve into the future (S < 1), with an equation $T \approx 13 - 16.5 \ln(S)$ Gy, with 13 roughly the y-intercept of the linear portion and 16.5 is Y_inf (Hubble time in far future).

This graphic plot goes forward into the future to S=0.1, when distances will be 10 times what they are today. And it starts around the emission of the CMB at S=1090. I'd like to generate a TABLE covering a similar span of cosmic evolution. I'll keep the upper limit at S=1090 but take the future to S=0.05, when distances will be 20 times today's. You'll see why. I'll use 29 steps to get down from 1090 to 1, which is what was suggested in Jorrie's post that I just quoted. Again I hope you will see why. It captures the maximum girth of the tear-drop shaped past lightcone.

I'll copy and paste the table I get in the next post. Here I'll simply quote some additional explanation. The maximum girth of the past lightcone comes when the distance at time of emission is EXACTLY EQUAL to the Hubble distance at that moment in the past. That is when the emission distance is increasing exactly at the speed of light. That is where the blue and green curves cross in Jorrie's figure. The table shows both emission distance (D_then) and Hubble distance, so you can see this equality, approximately, in the table. The Hubble distance is verbatim the same as Hubble time (T_Hub) if you just read years as lightyears.

Here's some additional explanation.

Part of what Jorrie was just talking about. I.e. stretch factor 2.63 and emission distance 5.8, has to do with the beautiful fact that past lightcones are TEAR-DROP SHAPE.

You can see that at the top level of the "figure 1" in my signature. That is what they look like when you measure in proper distance, the real distance that it actually was at the time, if you could have stopped the expansion process.

Other levels of the "figure 1" show conformal distance---what the distance to that same bit of matter would be today, not what it was back then. So the lightcone is not teardrop, it is some other shape.

the point of S=2.63 is that where the WIDEST bulge of the teardrop comes, in our past light cone. The largest girth. Farther back in time from then, the light cone PULLS IN. Of course that's because distances were smaller back then---and it is what gives it the teardrop or pear shape.

A rather beautiful thing happened around S=2.63 namely when galaxies emitted light then, that was destined to get here today for us to receive with telescopes, that light stayed at the same distance from us for a long time. Making barely if any progress. It stayed at distance 5.8, or more precisely according to the calculator, 5.798. Because its forward motion thru the surrounding space exactly canceled the rate at which the distance 5.798 was growing! So no net headway!

And then after a long time that distance 5.798 had slowed slightly and was not growing at the speed of light and the photons began to make headway towards us. The calculator will give an idea how long they took, all told, to get here. I think it was very nearly 10 billion years.

So you see in the preceding post Jorrie suggests putting 29 into the STEPS box, and also be sure to check the "exactly S=1" box so you get the exact present in your table. Then you will get, among much else, the S=2.62 line in the table, and that 5.8, and the time, what year it was etc.

The widest girth is at a crossing point in the figure which basically says the distance was growing at exactly c. You can see where the two curves cross. Blue and green. Blue for the emission distance, green for the Hubble distance (that distance which is growing at speed c.)

If you click on figure 1 in my signature you will also see a crossing of curves that marks this widest point on the teardrop lightcone. (In the top layer, the version drawn using proper distance. Other layers distort shape.)

 

[marcus]

So what I'll do is get the calculator, keep S_upper = 1090, put S_lower=0.05, put Steps=29.
I'ill check the box that makes the table include S=1 exactly, because I want the present moment in the table.
And I'll check the "copy&paste friendly" box, so I can paste the table that results here in this post.
If you do this yourself you get a lot more optional pop-up explanations, the units are listed. The cosmological parameters are explained in a heading to the table. Here's the link http://www.einsteins-theory-of-relativity-4engineers.com/TabCosmo5.html I'll essentially just copy and paste the numbers and leave the units out.

Copy&paste-friendly table? Check √ -----  Include S=1 (exact)? Check √

S	a	               T	T_Hub	       D_now	D_then	D_hor
1090.0	0.000917	0.000378	0.000637	45.731	0.042	0.056
856.422	0.001168	0.000566	0.000940	45.550	0.053	0.072
672.897	0.001486	0.000842	0.001381	45.341	0.067	0.091
528.701	0.001891	0.001247	0.002020	45.101	0.085	0.115
415.404	0.002407	0.001839	0.002944	44.825	0.108	0.146
326.387	0.003064	0.002700	0.004279	44.509	0.136	0.185
256.445	0.003899	0.003951	0.006205	44.150	0.172	0.234
201.491	0.004963	0.005761	0.008979	43.740	0.217	0.296
158.313	0.006317	0.008379	0.012973	43.275	0.273	0.373
124.388	0.008039	0.012159	0.018720	42.747	0.344	0.471
97.732	0.010232	0.017610	0.026985	42.149	0.431	0.593
76.789	0.013023	0.025465	0.038867	41.472	0.540	0.746
60.334	0.016574	0.036773	0.055945	40.706	0.675	0.937
47.405	0.021095	0.053047	0.080484	39.840	0.840	1.174
37.246	0.026848	0.076452	0.115738	38.861	1.043	1.468
29.265	0.034171	0.110103	0.166377	37.755	1.290	1.830
22.993	0.043491	0.158470	0.239106	36.507	1.588	2.275
18.066	0.055352	0.227971	0.343537	35.097	1.943	2.818
14.195	0.070449	0.327812	0.493442	33.506	2.360	3.474
11.153	0.089663	0.471192	0.708498	31.711	2.843	4.261
8.763	0.114117	0.677001	1.016667	29.686	3.388	5.192
6.885	0.145241	0.972188	1.457265	27.404	3.980	6.276
5.410	0.184854	1.394848	2.084258	24.837	4.591	7.513
4.250	0.235270	1.998124	2.968150	21.958	5.166	8.885
3.340	0.299437	2.853772	4.190977	18.748	5.614	10.347
2.624	0.381105	4.052600	5.822089	15.215	5.798	11.823
2.062	0.485047	5.694902	7.857010	11.408	5.534	13.201
1.620	0.617337	7.861899	10.128494	7.459	4.605	14.363
1.273	0.785708	10.571513	12.291156	3.574	2.808	15.228
1.000	1.000000	13.753303	13.999929	0.000	0.000	15.793
0.786	1.272738	17.277468	15.133799	-3.141	-3.998	16.121
0.715	1.399556	18.729987	15.440794	-4.230	-5.920	16.203
0.650	1.539011	20.208716	15.684266	-5.238	-8.061	16.267
0.591	1.692361	21.707838	15.875269	-6.167	-10.436	16.315
0.537	1.860992	23.223153	16.023472	-7.021	-13.066	16.351
0.489	2.046426	24.750714	16.137834	-7.804	-15.970	16.378
0.444	2.250336	26.287971	16.225336	-8.520	-19.174	16.398
0.404	2.474564	27.832518	16.292069	-9.175	-22.704	16.412
0.367	2.721136	29.382453	16.342940	-9.773	-26.593	16.421
0.334	2.992276	30.936767	16.381374	-10.318	-30.873	16.427
0.304	3.290433	32.494109	16.410600	-10.814	-35.583	16.430
0.276	3.618299	34.054029	16.432542	-11.266	-40.765	16.433
0.251	3.978834	35.615607	16.449246	-11.678	-46.465	16.449
0.229	4.375295	37.178725	16.461699	-12.053	-52.734	16.462
0.208	4.811259	38.742715	16.471229	-12.394	-59.629	16.471
0.189	5.290663	40.307651	16.478264	-12.704	-67.213	16.478
0.172	5.817837	41.873010	16.483706	-12.986	-75.552	16.484
0.156	6.397539	43.438976	16.487660	-13.243	-84.723	16.488
0.142	7.035005	45.005111	16.490781	-13.477	-94.808	16.491
0.129	7.735988	46.571662	16.492987	-13.689	-105.89	16.493
0.118	8.506820	48.138401	16.494627	-13.882	-118.09	16.495
0.107	9.354458	49.705116	16.496007	-14.058	-131.50	16.496
0.097	10.286558	51.272104	16.496902	-14.218	-146.25	16.497
0.088	11.311533	52.839007	16.497721	-14.363	-162.46	16.498
0.080	12.438640	54.406135	16.498195	-14.495	-180.30	16.498
0.073	13.678054	55.973144	16.498697	-14.615	-199.91	16.499
0.066	15.040966	57.540352	16.498931	-14.725	-221.47	16.499
0.060	16.539682	59.107420	16.499254	-14.824	-245.18	16.499
0.055	18.187733	60.674673	16.499353	-14.914	-271.26	16.499
0.050	20.000000	62.241776	16.499574	-14.997	-299.93	16.500

One reason I like going this far into the future, to S=0.05, is it shows both the Hubbletime and the Cosmic Event Horizon converging clearly to their longterm value of 16.5. One of the meanings of the cosmological constant is that it tells us what the Hubbletime will level out at eventually in the longterm future. What the longterm percentage expansion rate will be.
Currently it is 1/140 of a percent per million years. In the distant future it will be 1/165% per million years. The table shows that number emerging clearly. So it gives a concrete meaning to the cosmological constant.

 

[marcus]

Some extra explanation, probably not needed:
D_then, in the future, marks the limit of our future lightcone. It is the distance THEN, when the light arrives, of a galaxy we send a message to today.

D_horizon, now or at any given time in the future, is the distance at that time of the farthest galaxy which we could reach with a message sent at that time. For example D_horizon at present (namely 15.3 Gly) is the present distance of a galaxy which we could just barely reach with a message if we sent one today.

It is because of the cosmological constant (which we use in the form of a distance 16.5 Gly) that D_horizon converges to a finite distance which is, in fact, 16.5 Gly.
The cosmological constant originally appeared in the Einstein GR equation as a reciprocal area, which in our case would be (16.5 Gly)^-2. That is, one over the area which is the square of the horizon distance limit 16.5 Gly.

 

[marcus]

I want to try the experiment of using TabCosmo6 (version 6 of Jorrie's online tabular calculator) as a sort of time machine to see how our universe would have looked back in past e.g. S=2, or might look sometime in future e.g. S=0.5.

To go back in past, to S=2 when distances were half today's it's very simple: I get the calculator, put in Supper=2 and Steps=0
that gets a one-row table saying Time=5.9377 and Hubbletime=8.1357

So I am going back in time to year 5.94 billion (or 5.9377 to be more precise.)

The way I do this is I fetch the calculator again and put in 8.1357 instead of 14.0 for the present Hubbletime. And instead of 3280 for the era of matter-radiation equality I can put in 1640 (exactly half the stretch we see), but that adjustment is less important.

That's all that is needed, it is ready to go. So let's try making a table.
This is a table someone would have made back in year 5.94 billion. So from their standpoint the CMB stretch is 545 instead of 1090.

So let's put in Supper=545 and Slower=1 and keep Steps=10. You can see how it looks.
It gives the right age at recombination, year 378,000, and the right age of the universe 5.94 billion, as expected. But now let's change Slower to 0.5. This will take the table up to a time in that person's future which will correspond to OUR present day!
It should give the right time, namely year 13.75 billion.
And yes it does. Because it is the same universe, simply seen from the perspective of someone back in year 5.94 billion.
==========================
http://www.einsteins-theory-of-relativity-4engineers.com/TabCosmo6.html

We can also use version 6 to travel into the future and see how things look from there, e.g. from S=.5.
Same procedure. Fetch the calculator and put in Supper=.5 and Steps=0
That will tell us what year in the future we are going to and what the Hubbletime will be on that year.
It says 2.000000 24.380800 16.112811
namely distances will be twice today's, and it will be year 24.3808 billion and Hubbletime will be 16.1128 billion.
So I fetch the calculator again and put 16.1128 in place of 14.0
And instead of 3280 put 6560 (exactly twice) because for them in the future that will be the stretch they see for matter-radiation equality. Then it is ready to go, so let's see how it does.

I will ask it to tabulate from recombination Supper=2180 (the stretch from origin of CMB is now twice 1090) to Slower=1 (the presentday for these future people). Well, again it works!
The age comes out 24.38, and again recombination happens at year 378,000 agreeing exactly with the present-day figure the calculator normally gives.
Also if we put in enough steps it will come close to their S=2 which is our present day!
In fact with Steps=22 we hit 2.011 which is close, and the time and Hubbletime were
13.672240 13.965671 which round off to year 13.7 billion and 14.0 billion.
A one-shot calculation with S exactly equal to 2.000 would doubtless get the numbers exactly, but no need to bother, I think.

 

[marcus]

At present the critical matter density, for spatial flatness, is 0.23 nanopascal. This includes ordinary and dark matter together with electromagnetic radiation as a combined energy density. There is a physical constant which converts energy density (e.g. in pascal) into a square growth rate.

A growth rate is a reciprocal time, that is
1/(100 year) = 0.01/year = 1% per year
all represent the same instantaneous fractional rate of growth.
That is, what part of itself a quantity grows by, per unit of time.

A convenient growth rate scale for our purposes is percent per million years.

The physical constant I'm thinking of occurs in the Einstein GR equation and also in the Friedman equation of cosmology. It's value, in the appropriate terms is
6.195e-5 (% per million years)² per nanopascal.
You can see that if you multiplied this by an energy density in nanopascal you would get a squared growth rate.
Here's the bare bones arithmetic:
1/140^2 - 1/165^2 = (6.195e-5)*.23

You multiply this constant times 0.23 nanopascal (today's average energy density of ordinary and dark matter etc.) and you get the difference between two squared growth rates. Here's the arithmetic check---using expressions you can paste into google calculator.
((6.195e-5)*.23 + 1/165^2)^-.5 = 140

((6.195e-5)*.23 + 1/165^2)^.5 = 1/140

(6.195e-5)*.23 + 1/165^2 = 1/140^2
Try pasting this into the google window:
8 pi G/(3 c^2) in (percent per 10^6 year)^2 per nanopascal

It gives the value of the physical constant 8πG/(3c²)
in terms of square growth rate per nanopascal
namely in terms of (percent per million year)² per nanopascal

And if you paste the blue thing into the window you get 6.195e-5 in exactly those terms:
What the google calculator will in fact give you is 6.195e-5 (percent per 10^6 year)^2 per nanopascal.

It seems that one convenient way to write the Friedman equation of cosmology (with cosmological constant) is then as a relation between square growth rates on the left, and energy density on the right.

H² - H_∞² = 8πG/(3c²) ρ

where H(t) is the current growth rate of distance, H_∞ is the eventual growth rate it's tending towards namely 1/165% per million years, and ρ(t) is the current energy density of ordinary and dark matter.

 

[Jorrie]

At present the critical matter density, for spatial flatness, is 0.23 nanopascal. This includes ordinary and dark matter together with electromagnetic radiation as a combined energy density. There is a physical constant which converts energy density (e.g. in pascal) into a square growth rate.

Marcus, I don't follow you, because the 0.23 is a ratio between actual and critical density and nanopascal is a pressure. How do they connect?

 

[marcus]

Marcus, I don't follow you, because the 0.23 is a ratio between actual and critical density and nanopascal is a pressure. How do they connect?

I should have mentioned the pascal can serve as unit of energy density.
joule/cubic meter is equivalent to Newton/square meter

because if you take N/m² and multiply by meter and meter^-1
it does not change and you get Nm/m³ = J/m³

0.23 is not meant to be the dimensionless ratio of anything. It is intended to be an actual energy density of real stuff, namely DM and OM (ordinary matter) expressed as nanojoules per cubic meter.
It would break down as about 0.20 nanojoules per m³ for dark and
about 0.03 nanojoules per m³ for ordinary.
Or, with less rounding, something like 0.196 nJ/m³ for dark and 0.034 nJ/m³ for ordinary.

You may have more precise figures. I just calculated what the conventional ρ_crit is in nanojoules/m³ and multiplied by the conventional 27% which we have for OM+DM.

Here I'm seeing how it goes if you treat Lambda simply as the cosmological constant and the OM and DM densities as real (energy equivalent) densities.
In this way of looking at it, Lambda is just a curvature constant, not a density.

 

[marcus]

The new cosmological parameters from Planck, if they are sustained, seem likely to change some of our numbers.
http://arxiv.org/abs/1303.5076
NASA's newsletter about the Planck results announced today is:
http://science.nasa.gov/science-news/science-at-nasa/2013/21mar_cmb/
If the new results are upheld the estimated age would change to 13.8 from 13.7 billion years, according to the newsletter.
==quote==
Hubble's constant, is 67.15 plus or minus 1.2 kilometers/second/megaparsec... This is less than prior estimates derived from space telescopes, such as NASA's Spitzer and Hubble, using a different technique. The new estimate of dark matter content in the universe is 26.8 percent, up from 24 percent, while dark energy falls to 68.3 percent, down from 71.4 percent. Normal matter now is 4.9 percent, up from 4.6 percent.
==endquote==

If you put "1/(67.15 km/s per Mpc)" into google you get 14.56 billion years.

that means the current distance growth rate is around 1/145 or 1/146 of one percent per million years, instead of the 1/140 of one percent that some of us have been using. I am reluctant to change over until we hear more about this.
It makes the conventional critical "energy" density 0.76 nanopascal (including effect of vacuum curvature as "energy"..)

We just have to square the percentage and divide by 6.195e-5 to get the density in nanopascal. 1/145.6^2/6.195e-5 = 0.76.
So now if we believe Planck dark matter is 26.8% and ordinary matter is 4.9% for a total of 31.7% so the combined matter density, expressed in the same terms is
0.76*0.317=0.24 nanopascal for combined dark and ordinary matter density.

That leaves 0.52 nanopascal to be made up by the Lambda term
(6.195e-5*0.52)^-.5 = 176.
So the new H_∞ would be 1/176 percent per million years (down from 1/165.)
And the corresponding eventual Hubble time would be 17.6 billion years, up considerably from the 16.5 billion years we have been using.

Other readers may wish to work these things along different lines, but I am seeing how the numbers go if one keeps the Lambda term on the left as a geometric constant and does not treat it as a possibly fictional energy. I want to see how one would calculate the same stuff in that case.

 

[marcus]

The tabular calculator takes 3 main inputs
the now and eventual Hubble times, and the crossover S.

If I put in what I just calculated for the Hubble times and keep the crossover the same, I get the right age for the universe (according to NASA's letter and also page 11 table 2 of Planck report) and essentially the right parameters according to Planck.

So things are nicely consistent. Instead of 14.0 and 16.5 for the Hubble times, if I want to suit the Planck report, I just have to put in 14.56 and 17.6

Then the calculator gives me an age 13.83 billion years (NASA says 13.8, Planck's table 2 says 13.82) and it gives a Hubble parameter of 67.17 and a conventional matter fraction of 0.316.
This basically agrees with the "best fit" numbers in table 2 of the Planck report which were 67.11 and 0.317

As a nicety we could put the crossover S = 3400. Table 2 has something like 3402.

 

[marcus]

We have some new values for cosmic parameters from the Planck report and obviously they could be revised before settling down, so it could be premature to start using them in regular discussions. But...
I'm impressed by the Planck report and want to try changing over to the new numbers.
The three main parameters are the two key distance growth rates (or their reciprocals the two Hubble times) and the point in the past where you get matter/radiation equality--the new estimate for that is S=3400.

According to the Planck report the two key Hubble times are:
present 14.56 Gy
eventual 17.6 Gy

It always amazes me when I'm confronted by the elegance of the model. Just those three numbers and everything else follows. Hubble times are simply distance growth times--a way of specifying the distance growth rate which is the reciprocal of the interval of time. If we convert the two key Hubble times into their reciprocal growth rates we get:
present 1/145.6 % per million years
eventual 1/176 % per million years

These two numbers tell us what the density of matter must be in order to obtain the apparent near flatness of space. The average density of (ordinary and dark) matter can be stated in equivalent energy terms, namely joules per cubic meter:
matter density 0.24 nanojoules per cubic meter.

If we wish extra precision this would be 0.2403. It's quite close to 0.24 so I just quote that.

 

[marcus]

Recalling the essential content of post#44, there is a physical constant that occurs in the Einstein GR equation, and its simplified spin-off the Friedmann equation, that relates the difference of the two squared growth rates to the matter density. In matter, I include the small contribution from electromagnetic radiation, often without mentioning it.

H² - H_∞² = 8πG/(3c²) ρ

where H(t) is the current growth rate of distance, at present equal to 1/145.6% per million years. H_∞ is the eventual growth rate it's tending towards namely 1/176% per million years, and ρ(t) is the current energy density of matter (dark, ordinary and the small contribution from light) at present equal to 0.24 nanojoule per cubic meter, or 0.24 nanopascal.

The important physical constant, occurring in the Einstein GR equation, and also in the Friedman equation of cosmology, is 8πG/(3c²). Its value, in the appropriate terms, is
6.195e-5 (% per million years)² per nanopascal.

This can be seen by pasting into the google window this expression:
8 pi G/(3 c^2) in (percent per 10^6 year)^2 per nanopascal
This will make the google calculator give you the value of the physical constant 8πG/(3c²)
in terms of square growth rate per nanopascal
and it will say "6.195e-5 (percent per million year)² per nanopascal".

A moment's reflection shows that if you multiply this by an energy density in nanopascal you get a squared growth rate quantity. Here's the bare bones arithmetic:
1/145.6^2 - 1/176^2 = (6.195e-5)*.24
This is a condition specifying near spatial flatness. A squared growth rate corresponds to a negative space-time curvature. And any extra negative curvature beyond an inherent "vacuum curvature" level must be balanced by the positive contribution of matter.

Intuitively, far in the future when the density of matter (and light) is zero, both sides of this equation will be zero. The current square growth rate will simply equal the intrinsic vacuum curvature represented by the 1/176² term. So the difference on the left side of the equation will be zero.
But for the time being, the current square growth rate 1/145.6² is greater than the intrinsic curvature term---so to achieve spatial flatness, the excess must be balanced by the curvature generated by matter.

Anyway that's one way of looking at it and one way of presenting the new Planck numbers.

I'll illustrate with a sample section of the history of the universe using the new numbers and Jorrie's calculator.

BTW, apologies for using pascal (normally a pressure unit: Newton/m³) as shorthand for the equivalent SI unit of energy density: Joules/m³. The owners of SI metric terminology seem not to have thought necessary to give a separate name to the unit for when it is used as energy density, I guess because formally the two are equivalent.

 

[marcus]

Here's a sample cosmic history using the new Planck numbers. In Jorrie's cosmic tabulator I put
the Hubble expansion times 14.56 and 17.6, and the crossover S=3400, as per the Planck report.
Then to specify the dimensions of the table, I said
upper=45
lower=0.04
steps=20
and checked the "S = exactly 1" box.
This means the table goes back in past well before stars existed, when distances were 1/45 what they are today, and it goes out into future when distances are 25 times what they are today.
I also set all the columns to have 3-place precision. The often useful 6-place precision was not needed in this case.

$${\begin{array}{|c|c|c|c|c|c|c|}\hline Y_{now} (Gy) & Y_{inf} (Gy) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline14.56&17.6&3400&67.17&0.684&0.316\\ \hline\end{array}}$$ $${\begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&a=1/S&T (Gy)&T_{Hub}(Gy)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)\\ \hline45.000&0.022&0.056&0.085&39.362&0.875&1.247&0.153\\ \hline37.201&0.027&0.075&0.114&38.595&1.037&1.488&0.206\\ \hline30.753&0.033&0.100&0.151&37.750&1.228&1.772&0.277\\ \hline25.423&0.039&0.133&0.201&36.821&1.448&2.107&0.372\\ \hline21.017&0.048&0.178&0.268&35.798&1.703&2.500&0.498\\ \hline17.374&0.058&0.237&0.357&34.672&1.996&2.959&0.667\\ \hline14.363&0.070&0.316&0.475&33.434&2.328&3.494&0.893\\ \hline11.874&0.084&0.420&0.632&32.071&2.701&4.111&1.196\\ \hline9.816&0.102&0.559&0.841&30.573&3.115&4.821&1.599\\ \hline8.115&0.123&0.745&1.118&28.926&3.565&5.628&2.137\\ \hline6.708&0.149&0.991&1.485&27.115&4.042&6.538&2.855\\ \hline5.546&0.180&1.317&1.971&25.129&4.531&7.551&3.812\\ \hline4.584&0.218&1.751&2.610&22.951&5.006&8.659&5.086\\ \hline3.790&0.264&2.323&3.443&20.571&5.428&9.846&6.780\\ \hline3.133&0.319&3.076&4.516&17.982&5.740&11.084&9.028\\ \hline2.590&0.386&4.060&5.861&15.189&5.865&12.330&11.999\\ \hline2.141&0.467&5.325&7.485&12.215&5.705&13.526&15.903\\ \hline1.770&0.565&6.923&9.331&9.111&5.148&14.608&20.991\\ \hline1.463&0.683&8.883&11.256&5.963&4.075&15.518&27.544\\ \hline1.210&0.827&11.200&13.059&2.882&2.382&16.225&35.865\\ \hline1.000&1.000&13.834&14.560&0.000&0.000&16.730&46.281\\ \hline0.851&1.175&16.259&15.529&-2.253&-2.646&17.023&56.991\\ \hline0.725&1.380&18.819&16.232&-4.264&-5.883&17.220&69.718\\ \hline0.617&1.621&21.473&16.718&-6.040&-9.789&17.348&84.771\\ \hline0.525&1.904&24.191&17.040&-7.589&-14.446&17.429&102.522\\ \hline0.447&2.236&26.951&17.248&-8.928&-19.964&17.478&123.419\\ \hline0.381&2.627&29.739&17.380&-10.079&-26.473&17.507&147.994\\ \hline0.324&3.085&32.543&17.463&-11.065&-34.139&17.521&176.879\\ \hline0.276&3.624&35.358&17.515&-11.908&-43.154&17.526&210.819\\ \hline0.235&4.257&38.180&17.548&-12.627&-53.751&17.548&250.694\\ \hline0.200&5.000&41.006&17.568&-13.241&-66.203&17.568&297.535\\ \hline0.170&5.873&43.834&17.580&-13.763&-80.832&17.580&352.560\\ \hline0.145&6.899&46.664&17.588&-14.208&-98.017&17.588&417.194\\ \hline0.123&8.103&49.495&17.592&-14.587&-118.205&17.592&493.115\\ \hline0.105&9.518&52.327&17.595&-14.910&-141.917&17.595&582.294\\ \hline0.089&11.180&55.159&17.597&-15.185&-169.772&17.597&687.047\\ \hline0.076&13.133&57.991&17.598&-15.419&-202.490&17.598&810.091\\ \hline0.065&15.426&60.823&17.599&-15.618&-240.921&17.599&954.621\\ \hline0.055&18.119&63.656&17.599&-15.788&-286.064&17.599&1124.389\\ \hline0.047&21.283&66.488&17.600&-15.932&-339.089&17.600&1323.801\\ \hline0.040&25.000&69.321&17.600&-16.055&-401.373&17.600&1558.036\\ \hline\end{array}}$$
Time now (at S=1) or present age in billion years:13.834
'T' in billion years (Gy) and 'D' in billion light years (Gly)

 

[marcus]

One very beautiful thing about this table, as a sample segment of universe history, is that in the distant future one can see the cosmological constant Lambda emerging out of the fog, clearly, as a DISTANCE---a plainly visible concrete thing built into the universe's history.

By convention (going back to before 1920 with Einstein) a small positive Lambda corresponds to a slight negative spacetime curvature---that is a minus one over a large area quantity: the square of a length. So the naturally occurring Lambda constant in the Einstein equation is one over a length squared.

With the usual identification of time and distance, we can simply regard Lambda as a squared growth rate---one over an interval of time, squared. In other words the squared growth rate H_∞² in the equation a couple of posts back is an ALIAS for the cosmological constant Λ in the Einstein equation. (I'm neglecting a stray factor of 3.)

So when you look at the table and see the time quantity 17.6 Gy emerging at around year 60 billion in the future you are seeing a naked manfest appearance of the cosmological constant.

The same as when you see the distance 17.6 billion lightyears emerge, as the distance to the cosmological event horizon, eventually around year 60 billion in the future.
The reciprocal of that distance, squared, is again essentially the cosmological constant (indicating a slight constant negative space-time curvature) that Einstein wrote down in the equation which is now both our law of gravity and our law of geometry.

 

[marcus]

Another beautiful thing the cosmic history calculator shows you is the moment when the recession speed (of any chosen galaxy) stopped slowing down and began to pick up. It is an inflection point on the curve showing the distance to the galaxy. With WMAP numbers (pre-Planck mission 14, 16.5, 3280) this comes around year 7.3 billion.

Let's choose a galaxy which TODAY is at a distance equal to the Hubble radius: 14 billion lightyears.

The table is set to have 26 steps from S=1090 to exact present, and another 26 steps to S=.04.
You can see the minimum recession speed (rightmost column!) comes in the S=1.7 row, around year 7.3 billion.
You can also see that for the sample case we are tracking, where the distance today is 14 Gly, the current Hubble radius, the slowest recession speed ever attained is 0.8516 c. That is about 85% of the speed of light.

At present, because the galaxy is at Hubble radius, the recession speed is exactly c. And as you can also see from the table, in future it will continue to grow.

A galaxy at half the distance (now at 7 Gly instead of 14 Gly) would have a proportionally scaled recession speed history---just divide all the speeds by two! So knowing this one sample history let's us get the recession speeds for objects at other distances as well.

$${\begin{array}{|c|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{∞} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14&16.5&3280&69.86&0.72&0.28\\ \hline \end{array}}$$ $${\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)&v_{rec}\\ \hline 1090.000&0.000917&0.000378&0.000637&45.731&0.042&0.056&0.001&20.1636\\ \hline 832.918&0.001201&0.000592&0.000983&45.527&0.055&0.074&0.001&17.0999\\ \hline 636.471&0.001571&0.000922&0.001508&45.289&0.071&0.096&0.002&14.5867\\ \hline 486.356&0.002056&0.001428&0.002302&45.009&0.093&0.125&0.003&12.5044\\ \hline 371.647&0.002691&0.002197&0.003500&44.684&0.120&0.163&0.005&10.7630\\ \hline 283.992&0.003521&0.003365&0.005304&44.307&0.156&0.212&0.008&9.2950\\ \hline 217.011&0.004608&0.005131&0.008015&43.872&0.202&0.275&0.013&8.0485\\ \hline 165.828&0.006030&0.007798&0.012088&43.369&0.262&0.357&0.020&6.9840\\ \hline 126.717&0.007892&0.011817&0.018200&42.790&0.338&0.462&0.031&6.0704\\ \hline 96.830&0.010327&0.017862&0.027367&42.124&0.435&0.598&0.047&5.2831\\ \hline 73.992&0.013515&0.026948&0.041109&41.360&0.559&0.773&0.072&4.6026\\ \hline 56.541&0.017686&0.040590&0.061703&40.483&0.716&0.996&0.109&4.0129\\ \hline 43.205&0.023145&0.061058&0.092556&39.477&0.914&1.280&0.167&3.5009\\ \hline 33.015&0.030289&0.091754&0.138771&38.325&1.161&1.640&0.253&3.0557\\ \hline 25.228&0.039638&0.137768&0.207983&37.005&1.467&2.093&0.383&2.6682\\ \hline 19.278&0.051872&0.206718&0.311611&35.494&1.841&2.661&0.580&2.3305\\ \hline 14.731&0.067883&0.310005&0.466715&33.764&2.292&3.365&0.876&2.0363\\ \hline 11.257&0.088835&0.464670&0.698717&31.784&2.824&4.228&1.323&1.7800\\ \hline 8.602&0.116254&0.696135&1.045272&29.520&3.432&5.269&1.994&1.5571\\ \hline 6.573&0.152136&1.042148&1.561411&26.934&4.098&6.502&3.003&1.3641\\ \hline 5.023&0.199093&1.558281&2.325166&23.985&4.775&7.922&4.517&1.1988\\ \hline 3.838&0.260543&2.324459&3.439363&20.641&5.378&9.496&6.782&1.0605\\ \hline 2.933&0.340960&3.450250&5.016065&16.884&5.757&11.146&10.156&0.9516\\ \hline 2.241&0.446198&5.070303&7.113058&12.751&5.689&12.742&15.136&0.8782\\ \hline 1.713&0.583918&7.312958&9.599448&8.373&4.889&14.119&22.363&0.8516\\ \hline 1.309&0.764145&10.232782&12.059647&4.011&3.065&15.144&32.599&0.8871\\ \hline 1.000&1.000000&13.753303&13.999929&0.000&0.000&15.793&46.686&1.0000\\ \hline 0.764&1.308652&17.700005&15.230903&-3.469&-4.539&16.147&65.616&1.2029\\ \hline 0.682&1.465878&19.447858&15.566734&-4.731&-6.935&16.236&75.350&1.3183\\ \hline 0.609&1.641994&21.229081&15.819561&-5.879&-9.654&16.301&86.289&1.4531\\ \hline 0.544&1.839269&23.035135&16.007122&-6.919&-12.726&16.348&98.568&1.6086\\ \hline 0.485&2.060245&24.859344&16.144845&-7.857&-16.187&16.380&112.342&1.7865\\ \hline 0.433&2.307770&26.697095&16.244907&-8.700&-20.077&16.402&127.785&1.9889\\ \hline 0.387&2.585034&28.544549&16.317231&-9.457&-24.446&16.416&145.094&2.2179\\ \hline 0.345&2.895609&30.399001&16.369270&-10.135&-29.346&16.425&164.489&2.4765\\ \hline 0.308&3.243498&32.258319&16.406749&-10.742&-34.841&16.430&186.221&2.7677\\ \hline 0.275&3.633183&34.121403&16.433445&-11.285&-41.000&16.433&210.567&3.0952\\ \hline 0.246&4.069687&35.987064&16.452507&-11.770&-47.901&16.453&237.840&3.4630\\ \hline 0.219&4.558633&37.854565&16.466097&-12.204&-55.634&16.466&268.393&3.8759\\ \hline 0.196&5.106324&39.723214&16.475939&-12.592&-64.297&16.476&302.617&4.3390\\ \hline 0.175&5.719816&41.592963&16.482824&-12.938&-74.001&16.483&340.955&4.8582\\ \hline 0.156&6.407015&43.463378&16.487715&-13.247&-84.873&16.488&383.899&5.4403\\ \hline 0.139&7.176777&45.334268&16.491186&-13.523&-97.051&16.491&432.003&6.0926\\ \hline 0.124&8.039020&47.205331&16.493809&-13.769&-110.692&16.494&485.887&6.8235\\ \hline 0.111&9.004857&49.076799&16.495546&-13.989&-125.973&16.496&546.245&7.6425\\ \hline 0.099&10.086732&50.948438&16.496771&-14.186&-143.090&16.497&613.854&8.5601\\ \hline 0.089&11.298588&52.820200&16.497630&-14.361&-162.263&16.498&689.587&9.5881\\ \hline 0.079&12.656041&54.691883&16.498394&-14.518&-183.740&16.498&774.418&10.7395\\ \hline 0.071&14.176583&56.563793&16.498808&-14.658&-207.797&16.499&869.442&12.0295\\ \hline 0.063&15.879808&58.435746&16.499091&-14.783&-234.745&16.499&975.882&13.4745\\ \hline 0.056&17.787665&60.307731&16.499279&-14.894&-264.931&16.499&1095.110&15.0932\\ \hline 0.050&19.924739&62.179573&16.499566&-14.994&-298.742&16.500&1228.663&16.9063\\ \hline 0.045&22.318568&64.051596&16.499641&-15.082&-336.617&16.500&1378.261&18.9374\\ \hline 0.040&25.000000&65.923630&16.499682&-15.162&-379.041&16.500&1545.833&21.2125\\ \hline \end{array}}$$Time now (at S=1) or present age in billion years: 13.753301
'T' in billion years (Gy) and 'D' in billion light years (Gly), sample recession speed history of matter now at distance R₀, shown as multiples of the speed of light
====

 

[marcus]

Using model parameters from the recent Planck mission report we get nearly the same recession speed history as above.
From Planck, combined with earlier data, we get 14.4 Gly, 17.3 Gly, and 3400. Plugging these parameters into the calculator we get that the minimum recession speed comes at S=1.652 and year 7.592 billion. For a galaxy which is now at current Hubble radius R_o = 14.4 Gly from us, the minimum recession speed is 0.87258c .

So 87% of the speed of light, instead of 85% (as found with earlier model parameters). I think the difference is mainly due to the longer Hubble radius 14.4 instead of 14.0. The representative galaxy we choose to track is slightly more distant, so its recession speeds are slightly higher throughout history, including the minimum.

The minimum is attained somewhat later, namely year 7.6 billion instead of year 7.3 billion which we found in preceding post using 2010 WMAP parameters.

One thing that is easy to do with the table calculator is see what happens when you vary parameters slightly. You can find for instance that the increasing the eventual Hubble radius R_∞ (keeping the other two the same) will make the minimum speed come later.
That makes sense--it delays the onset of "accelerated expansion". A cosmological constant of zero would correspond to infinite Hubble radius, and the expansion speed would continue declining indefinitely and never bottom out. So the longer R_∞ is, the longer you have to wait for acceleration to occur. Accordingly, we see the year of the minimum change from 7.3 to 7.6 billion when we adopt Planck mission numbers and increase R_∞ from 16.5 to 17.3 Gly.

 

[marcus]

There might eventually be a "learner's manual" to go with Jorrie's calculator so I'll experiment with a few cosmic history tables that one can find things into point out and discuss. Here is one that shows the "deja vu" epoch. An earlier time when any galaxy would have the same recession speed that it does right now. This comes around year 3.3 billion. The table also shows a few other points of interest. It runs from S=10 around the time the the first galaxies formed, up to present S=1 and then on to S=0.1 when distances will be ten times what they are now.
I used Planck 2013 model parameters and specified 17 steps from start to present.$${\scriptsize \begin{array}{|c|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{∞} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline14.4&17.3&3400&67.92&0.693&0.307\\ \hline \end{array}}$$'T' in billion years (Gy) and 'D' in billion light years (Gly), a sample recession speed history of matter now at distance R_o is shown in multiples of the speed of light.$${\scriptsize \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)&v_{rec}sample\\ \hline 10.000&0.100&0.545&0.820&30.684&3.068&4.717&1.558&1.76\\ \hline 8.733&0.115&0.668&1.004&29.536&3.382&5.270&1.916&1.64\\ \hline 7.627&0.131&0.819&1.229&28.307&3.711&5.873&2.354&1.54\\ \hline 6.661&0.150&1.004&1.504&26.994&4.053&6.528&2.893&1.44\\ \hline 5.817&0.172&1.229&1.840&25.591&4.399&7.234&3.554&1.35\\ \hline 5.080&0.197&1.505&2.249&24.093&4.743&7.988&4.364&1.26\\ \hline 4.437&0.225&1.842&2.744&22.495&5.070&8.786&5.357&1.18\\ \hline 3.875&0.258&2.253&3.341&20.794&5.367&9.622&6.573&1.11\\ \hline 3.384&0.296&2.753&4.056&18.988&5.611&10.484&8.061&1.05\\ \hline 2.955&0.338&3.358&4.903&17.077&5.779&11.357&9.877&0.99\\ \hline 2.581&0.387&4.088&5.891&15.065&5.837&12.225&12.089&0.95\\ \hline 2.254&0.444&4.960&7.017&12.963&5.751&13.066&14.775&0.91\\ \hline 1.968&0.508&5.994&8.264&10.788&5.481&13.856&18.023&0.89\\ \hline 1.719&0.582&7.203&9.592&8.567&4.983&14.574&21.929&0.87\\ \hline 1.501&0.666&8.593&10.941&6.334&4.219&15.201&26.597&0.88\\ \hline 1.311&0.763&10.164&12.235&4.132&3.151&15.726&32.134&0.90\\ \hline 1.145&0.873&11.902&13.405&2.002&1.749&16.147&38.655&0.94\\ \hline 1.000&1.000&13.787&14.400&0.000&0.000&16.472&46.279&1.00\\ \hline 0.873&1.145&15.794&15.201&-1.890&-2.164&16.714&55.139&1.08\\ \hline 0.763&1.311&17.896&15.814&-3.607&-4.729&16.888&65.388&1.19\\ \hline 0.666&1.501&20.071&16.267&-5.157&-7.743&17.010&77.200&1.33\\ \hline 0.582&1.719&22.297&16.591&-6.544&-11.249&17.093&90.781&1.49\\ \hline 0.508&1.968&24.561&16.818&-7.775&-15.305&17.149&106.372&1.69\\ \hline 0.444&2.254&26.850&16.974&-8.863&-19.976&17.185&124.252&1.91\\ \hline 0.387&2.581&29.157&17.081&-9.820&-25.343&17.207&144.745&2.18\\ \hline 0.338&2.955&31.475&17.153&-10.660&-31.502&17.220&168.223&2.48\\ \hline 0.296&3.384&33.802&17.202&-11.396&-38.563&17.226&195.115&2.83\\ \hline 0.258&3.875&36.135&17.234&-12.041&-46.654&17.234&225.913&3.24\\ \hline 0.225&4.437&38.470&17.256&-12.605&-55.923&17.256&261.183&3.70\\ \hline 0.197&5.080&40.809&17.271&-13.098&-66.538&17.271&301.571&4.24\\ \hline 0.172&5.817&43.149&17.280&-13.528&-78.695&17.280&347.819&4.85\\ \hline 0.150&6.661&45.490&17.287&-13.905&-92.617&17.287&400.776&5.55\\ \hline 0.131&7.627&47.832&17.291&-14.233&-108.558&17.291&461.415&6.35\\ \hline 0.115&8.733&50.174&17.294&-14.521&-126.813&17.294&530.850&7.27\\ \hline 0.100&10.000&52.516&17.296&-14.772&-147.715&17.296&610.357&8.33\\ \hline \end{array}}$$
The sample galaxy's present-day recession speed is 1c, the speed of light. Deja vu is at S=3.00,when the galaxy was also receding at the speed of light. The table comes close enough (S=2.955) so that the speed in that row of the table is 0.99c.
Minimum speed occurs around S=1.7. Looking at that row of the table, one can see that for the sample galaxy we've chosen the slowest it ever is, in the whole of cosmic history, is 0.87c, 87% of the speed of light.

In its D_then column the table also shows the radius of the past lightcone. It is the distance of something we are now getting light from at the time it emitted the light. You can see by scanning down the D_then column that the greatest distance at the time of emission is 5.8 billion light years. An emitter at this maximum remove is receding exactly at speed c, so that the light we are receiving from it at first "stood still" (could not close the distance between us) but later began to make headway. D_then coincides with Hubble radius R at that moment in time, as the table also shows.

 

[marcus]

Today I happened to get curious about early times, not the first second or few minutes of the cosmos but something simpler to picture, like year 2000 from the start of expansion. So I put in S=20000.
Ooops, have to go to supper. back later, here's the output for that stretch$${\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$$ $${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.00005&20000.0&0.00000187&0.00000350&46.177&0.002309&3.21&659.18\\ \hline \end{array}}$$

You can see it is year 1,870. Just a bit before year 2000. I'll think about what it says conditions were like, after supper.

Part of this is just learning to read off from the table, and get the decimal point in the right place. You know what the temperature of of the CMB is today, around 2.76 kelvin. To get the temperature of radiation back then I guess you just multiply by 20000, or by whatever S is at the time. So 5.5 x 10⁴ kelvin---i.e. around 55,000 kelvin.

And the cube of S is 8 x 10¹². So the density of matter was 8 trillion times what it is today. But that isn't all that much because on average it is so scarce today. amounts to only about 0.23 nanojoule per cubic meter. energy equivalent, including dark matter which is the bulk of it.

So back then, in year 1870, a cubic meter contained 1840 joules worth of matter
1840 joules/c^2 into Google gives: 2 x 10^-11 grams. I can hardly believe it is so little!
Well that is what it seems to be.

 

[Jorrie]

$${\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$$ $${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.00005&20000.0&0.00000187&0.00000350&46.177&0.002309&3.21&659.18\\ \hline \end{array}}$$
Part of this is just learning to read off from the table, and get the decimal point in the right place. ...

I find it useful to set the decimals to 9 for such small values, because then the digits represent years or light years. For slightly larger minima, six decimal digits obviously represent My and so on. Sadly, one can't change it halfway through a long table...

 

[Neko]

Long ago in a galaxy far far ...

At what cosmological distance can we be confident the source of the light we see exists today? My son just told me it's spooky action at a distance.

 

[Jorrie]

At what cosmological distance can we be confident the source of the light we see exists today? My son just told me it's spooky action at a distance.

Galaxies are the farthest sources that we are pretty confident that they still exist today, because they are (sort-of) regenerating stars from the gas that that they lock up. The farthest confirmed one that I know of is MACS0647-JD at redshift of 10.9, meaning light took 13.3 billion years to reach us. Due to cosmic expansion, MACS0647-JD must be some 32 billion light years away today.

Potentially farther galaxies are continuously discovered, but it takes some time for the redshifts to be confirmed by other resources.

 

[mfb]

My son just told me it's spooky action at a distance.

This term is used in a completely different context, and has nothing to do with light of old galaxies.

 

[Neko]

Spooky action at a distance

This term is used in a completely different context, and has nothing to do with light of old galaxies.

Jorrie:

Thank you. I understand the term is generally used in a QP context. My son has a sense of humor. He was transcending parsecs and Hubble and red shift. Do you have any thoughts on the existential question?

Neko

 

[mfb]

No philosophy here, please, that usually leads to nothing.

 

[bobie]

It's great to have a Cosmic Event Horizon column!

It is regrettable that Jorrie is inactive at the moment. I hope you can help , marcus.

According to the calculator, the horizon now is just a liitle greater than the radius (14.4 vs 16.4), why so? we know that the ant always reaches its goal, however distant, even if the V_Erubber espansion rate is 100 000 times greater than its own speed v_ant.
Why such a great difference here? Are the formulas different? The conditions seem nearly the same,
nay, much better since recession speed V_EU ≈ V_light.

The link says the formula for D_hor is 1/S ∫^S₀ dS/H,
where can I find the original formula and an explanation?

Thanks

 

[Jorrie]

According to the calculator, the horizon now is just a liitle greater than the radius (14.4 vs 16.4), why so? we know that the ant always reaches its goal, however distant, even if the V_Erubber espansion rate is 100 000 times greater than its own speed v_ant.

The "ant always reaches its goal" only in the case of coasting or decelerating models, because they have an infinite cosmic event horizon radius. In an accelerating model, the Hubble radius (1/H0) always tends towards the event horizon radius as time goes on.

Why such a great difference here? Are the formulas different? The conditions seem nearly the same, nay, much better since recession speed V_EU ≈ V_light.

I do not quite understand your question, but the relationship between the time light took to reach us (observably universe), the Hubble radius and the event horizon is graphically shown in an attachment (graph from calculator)

 

[bobie]

The "ant always reaches its goal" only in the case of coasting or decelerating models, because they have an infinite cosmic event horizon radius. In an accelerating model, the Hubble radius (1/H0) always tends towards the event horizon radius as time goes on.

Hi Jorrie, I am so glad you replied! I have a few questions nobody could answer.
Can you explain the difference between a stretching rubber balloon and the stretching space?
If you refer to the growing rate of expansion, that is really microscopic, even if it did fluctuate it is on the average ≈1/T₀, we can easily consider it stable and V_e ≈ C.
If you refer to other factor please expand on it.

I do not quite understand your question, but the relationship between the time light took to reach us (observably universe), the Hubble radius and the event horizon is graphically shown in an attachment (graph from calculator)

I was referring to the formula of the rubber band,which is so different from the one you are using, but probably when you explain the difference it will all be clarified.

One more thing:
what is and what is the formula for V now/then? In the link I did not find them and nobody could tell me.
For example for S = 1090 V_then is 3.15c does it mean the (apparent) recession speed is 3.15c?
and V_now is 66.18, what does it represent? and what is the formula, $$1090 = \sqrt{\frac{c+v}{c-v}}$$
Thanks a lot, again

 

[Jorrie]

Bobie, the expanding/stretching balloon is no more than a simple analogy to make a small part of cosmology (most importantly, the distance/redshift relationship) easier to understand - space is nothing like rubber! Stick pennies onto a partially inflated balloon and when it is blown up further, the distances between the pennies change according to Hubble's law. That's all there is to it.

If you refer to the growing rate of expansion, that is really microscopic, even if it did fluctuate it is on the average ≈1/T_{0 ...}

No, I think you are confusing the growth of the apparent radius of the observable universe (which grows at c in appropriate units) with the growth of the distance between remote galaxies. The observable universe depends solely on the time since the BB and its radius is not a distance in the true sense of the word. Expansion rates refer to the change of the proper distance between galaxies over time.*

... what is and what is the formula for V now/then? In the link I did not find them and nobody could tell me. For example for S = 1090 V_then is 3.15c does it mean the (apparent) recession speed is 3.15c?

The expansion rate changes drastically over the history of the universe and there is no simple "V now/then" formula, but the V's are readily calculable from the Hubble parameter H against expansion factor and time. Refer to this Wiki that Markus, Mordred and I worked on some time ago.

-J

* See the definition of D_now in the "Show columns definition and selection" (hover over the question mark) in LightCone 7.

 

[Jorrie]

LightCone7zeit has undergone a minor enhancement of the charting function.

The Chart Options area is opened/closed from the main screen. It allows more customizable charts, limited only by what Google Charts will allow.

It only works in the 'zeit version' of LightCone7 at present. If deemed useful, the standard 'billion years version' could also be upgraded with this functionality in future.

--
Regards
Jorrie

 

[Bandersnatch]

Hi, @Jorrie
We've just had a thread in which a poster wanted to see the evolution of Hubble constant with time. @marcus posted a graph from the 7zeit calculator, but those units the calc uses are not the easiest to comprehend for a neophyte. On the other hand, I've noticed that there is no H column available for display in your other calc (light cone 7). There is the reciprocal (Hubble radius) there, so adding Hubble constant should be relatively easy (not that I know anything about programming).
Do you think you could add such a column as an optional selection for further reference in the non-zeit calculator?

 

[Jorrie]

Hi, @Jorrie
We've just had a thread in which a poster wanted to see the evolution of Hubble constant with time. @marcus posted a graph from the 7zeit calculator, but those units the calc uses are not the easiest to comprehend for a neophyte. On the other hand, I've noticed that there is no H column available for display in your other calc (light cone 7). There is the reciprocal (Hubble radius) there, so adding Hubble constant should be relatively easy (not that I know anything about programming).
Do you think you could add such a column as an optional selection for further reference in the non-zeit calculator?

Yes, it is easy and I already have a draft version of the 'standard' LightCone7 with the H column. The reason for not having released it yet is that I have not decided on the units for graphing it - it has rather awkward units; either it is way smaller than 1 (presently 0.069/Gy), or it is way larger than 1 (68 km/s/Mpc). I was thinking about making it H/Ho, which will pitch it around unity for the present epoch, but then it is still pretty small compared to R, T etc.

In tabular form it obviously does not matter too much, but then one of LightCone's greatest features is the charting...

 

[marcus]

My thoughts might not be pertinent, but I'm glad to see you both in this thread. Here's how I might explain to a neophyte.
H(t) is an instantaneous speed-to-size ratio.
It is what you multiply a distance of size D by to get the speed that distance is expanding.

So the clearest way to express H₀ = H(now) is $\frac{1}{14.4 Gy}$

If you take the distance 14.4 Gly and multiply by that, you get $\frac{14.4 Gly}{14.4 Gy}$ namely the speed of light, which is the right thing.

If you take any other largescale distance and multiply by $\frac{1}{14.4 Gy}$ you get the speed that distance is currently expanding.