Could a cosmo tabulator give hands-on experience with cosmic Event Horizon?

[marcus]

Cosmo calculators and tabulators a primarily about the PAST expansion history and they give learners hands-on understanding by being able to vary the model parameters and see change. That's good. I sometimes notice a difference here at PF between how posters with mainly verbal understanding think and how those do who also have some of the quantitative feel you get from playing with an equation model of the cosmos or with one of the online interactive models.

However the murky talk about "dark energy" and all that is essentially about a FUTURE leveling off of the declining Hubble expansion rate, to about 60 km/s per Mpc. Or if you like about the plateauing of its reciprocal, the Hubble time, at about 16.3 billion years. And this adds a very interesting feature to our universe, which has to do with the FUTURE--namely the CEH (cosmic event horizon).

I'm not sure of its exact estimate right now, somewhere between 15 and 16 Gly. It is gradually converging to 16.3 Gly. So let's say it is now "around 16" billion lightyears.

This is a very interesting thing. If a galaxy is NOW less than ~16 Gly from us and we send them some light TODAY the light will eventually get there. Though if the galaxy is very close to the 16 Gly limit it could take a very very long time to reach them. The closer it is to the limit the longer it will take, because given the expansion of distances going on it can just barely make it, traveling at the speed of light.

However if the galaxy is NOW beyond the ~16 limit and we send them a flash today it will never get there.

And it works both ways. We see lots of galaxies out there which are farther than 16 Gly. In fact most of the galaxies you can see with a telescope are farther than that. But if something happens TODAY in one of those galaxies we will never see it. They are beyond the current event horizon and they cannot now send us a message that will ever reach us.

So that is an interesting feature of the universe which basically has to do with this plateauing of the Hubbletime at 16.3 Gly and involves future time. Can a cosmo tabulator give us hands-on experience with future expansion as well as experience with our universe's past expansion history?

You'd have to be able to play around with the Hubbletime parameters (ie. with the two expansion rate parameters) and see how the distances change. Namely the distance NOW to the galaxy we want to send light to, and the distance THEN to the galaxy when the light finally gets there (despite all the expansion that will have been happening).
Does anyone know of an online resource like this, or have any ideas? It could be helpful for people learning about cosmology, and kind of neat actually

 

[Drakkith]

I could see a simple flash animation with some slider bars for the variables such as distance being useful.

 

[marcus]

I could see a simple flash animation with some slider bars for the variables such as distance being useful.

It's a suggestive idea. It got me started playing around with the two main parameters of the mode---the two Hubbletimes (call them for lack of any neater notation Y_now and Y_∞). Reciprocal growth rates, now and at infinity.
If you multiply them by log(2) you get what would be the distance doubling times at that constant rate of distance growth.
If you divide a distance by the current Y you get the speed it is increasing. (distance divided by time equals speed and this is the time to divide by).

So the Hubbletime at any epoch is a handy package of information about expansion, and the two Hubbletimes--now and at infinity--are the two crucial parameters that we get everything else from (given the flat Friedman equation and the observed proportion of light to matter.)

I tried varying the two Hubble times and noticed that they have opposite effects on an interesting distance: the farthest distance anything we can see was when it emitted the light.

We never see anything that was farther than 5.8 billion lightyears when it emitted the light and this maximum "as-we-see-it" was achieved in the S=2.64 era (when distances were 1/2.64 what they are today).

You can see the max stand out clearly if you use the calculator with upper limit S=3.0
lower limit S=2.5 and step=0.01
It's quite pronounced, and a bit curious.

So I was prompted by your post to try varying the two Hubbletimes 13.9 and 16.3 Gly to see how that would change.

INCREASING Y_now makes this interesting max distance bigger and come sooner.
Decreasing Y_now makes it occur later and smaller.

But changing the longterm limit Hubbletime has the opposite effect.

INCREASING Y_∞ makes this max as-we-see-it distance later and smaller.
Decreasing Y_∞ makes it earlier and bigger.

I didn't try varying both at once but it's pretty clear that if you bring 13.9 and 16.3 closer together by increasing the former and decreasing the latter both at once, you get that maximum distance earlier (i.e. with a larger S) and bigger.
And increasing the Hubbletime separation by simultaneously decreasing 13.9 and increasing 16.3 would have the opposite effect.

Anyway Drakkith, this is an example of using the online cosmic model as if it had slider bars.

You just set:
upper S=3.0
lower S=2.5
step=0.01
and play around with the two Hubbletimes, varying Y_now and Y_∞.

this maximum as-we-see-it distance still intrigues me and I'd like to get more intuition about it. It always occurs just when the THEN distance is exactly equal to the Hubble distance which is cY---that is the then Hubble time multiplied by the speed of light.

I'd like to understand better why that happens.

 

[Jorrie]

I didn't try varying both at once but it's pretty clear that if you bring 13.9 and 16.3 closer together by increasing the former and decreasing the latter both at once, you get that maximum distance earlier (i.e. with a larger S) and bigger.
And increasing the Hubbletime separation by simultaneously decreasing 13.9 and increasing 16.3 would have the opposite effect.

When you set the two Ys closer together, you effectively increase the cosmological constant relative to matter and radiation. Accelerating expansion will hence start earlier in time. Will think about other puzzles tomorrow - past mid-nite here...

 

[marcus]

The capability of the new A20 version to go into the future is GREAT!

I tried a sample and in confirms what the toy model says that the time when distances will be 100 times what they are today will be about 88 billion years in the future.
That is the S=0.01 era. The S number is always a comparison of OUR lengths and distances with those of the other era.

So S=10 is the era in the past when distances were 1/10 those of today.
S=0.1 is the era in the future when distances will be 1/.1 of those today (in other words 10 times distances now.)

In effect the S tells what our present distances look like compared with theirs (whether they are in the past or in the future.) I'm telling myself this to get used to the terminology.

I think it is really great to be able to look into the future and actually SEE THE COSMIC EVENT HORIZON looming up in front of us at around 15.5 to 16 billion light years.

this is the distance limit on the galaxies NOW that we can expect to send light to that they will eventually receive. The calculator is exploring our forward light cone, by numerical integration, in an expanding universe. I can see D_now beginning to converge.

 

[Jorrie]

...
I think it is really great to be able to look into the future and actually SEE THE COSMIC EVENT HORIZON looming up in front of us at around 15.5 to 16 billion light years.

this is the distance limit on the galaxies NOW that we can expect to send light to that they will eventually receive. The calculator is exploring our forward light cone, by numerical integration, in an expanding universe. I can see D_now beginning to converge.

I have used Tamara's equation for comoving event horizon distance:
$$\chi_c(t) = c \int_{t}^{t_{end}} \frac{dt'}{R(t')}$$
to approximate the proper CEH and it gives a present value of 15.64 Gly.

As far as can see, in our present calculator parlance, the proper distance to the horizon will translate to:
$$D_{CEH} = \frac{1}{S} \int_{0}^{S}{\frac{dS}{H}}$$

where $H = H_0 \sqrt{\Omega_\Lambda + \Omega_m S^3 (1+S/S_{eq})}$,
as per this post.

I guess this could perhaps easily be incorporated into a calculator.

 

[George Jones]

We never see anything that was farther than 5.8 billion lightyears when it emitted the light

...

this maximum as-we-see-it distance still intrigues me and I'd like to get more intuition about it. It always occurs just when the THEN distance is exactly equal to the Hubble distance which is cY---that is the then Hubble time multiplied by the speed of light.

I'd like to understand better why that happens.

There's something curious here. It is a different "horizon" that we don't normally hear about.
This 5.8 billion lightyears is the maximum distance we can see things in the sense that it is the farthest away they could be at time the light was emitted.

It is the distance THEN maximum. Small angular size corresponds to far away at the time of emission. And smallest angular size necessarily has to correspond to greatest THEN distance

...

Maybe this figure can help us understand why 5.8 billion ly is the farthest then-distance we can see.

If you look at the top graph of that same figure, which has proper distance, you notice that the lightcone has a teardrop shape, there is a point where it is fattest, and its tangent is vertical. that is the place where it is widest and the point we are talking about. Its diameter is 5.8 billion ly there. It is also where the Hubble radius crosses. I think you can see that in the figure. It is interesting. There is also an intersection about the same time level, between the particle horizon and the cosmic event horizon.

In case anyone hasn't read the original Lineweaver article, and wants to:
http://arxiv.org/abs/astro-ph/0305179
The figure is on page 6
The same figure is used in the Lineweaver-Davis article that is often referred to:
Look on page 3 of http://arxiv.org/pdf/astro-ph/0310808.pdf
The graphic quality is better there---plots show up larger plus there's explanatory text as well!

People always use the word "teardrop" to describe the shape of the lightcone plotted in proper distance. I would rather say an entirely convex pearshape, like this Anjou pear:tongue2:
http://carrotsareorange.com/wp-content/uploads/2010/05/pear-anjou.jpg
The "equator" we are referring to is analogous to a belt around the widest part of the pear.

Regret to say: no helpful ideas about the exposition at the moment. Maybe some will come.

marcus, I think that you have explained the intuition behind this in other, related posts about our particular expanding and accelerating universe.

The top panel of Figure 3 is an even better, expanded diagram that shows the Hubble sphere crossing our past lightcone at the lightcone's maximum "bulge" of 5.8 billion ly. Now move to pages 18 and 19.

Supposes that a galaxy on our past lightcone emits a photon towards us when it: 1) is a distance of 4 billion ly from us; 2) is at time earlier than the bulge. The photon moves up our past lightcone until it reaches us. Differentiating equation (16) gives equations (19) and (20) along the worldline (our past lightcone) of the photon, with $v_{pec} = -c$ (negative, since the photon is directed towards us). Consequently, for the photon, all along the photon's worldline,

$$v_{tot} = v_{rec} - c,$$

where (using the paper's notation)

$$v_{rec} = \dot{R} \chi = \frac{\dot{R}}{R}R \chi = HD$$

Initially, $v_{rec}$ is greater than $c$, so $v_{tot}$ is positive, and the photon moves away from us, i.e., $D$ increases. In our universe, the Hubble parameter $H$ decreases as time increases. For this photon, $H$ decreases faster than $D$ increases, so $v_{rec} = HD$ and $v_{tot} = v_{rec} - c$ both decrease. The photon's worldline curves upwards until $v_{rec}$ has decreased to the value of $c$, at which time 1) the photon is on the Hubble sphere by definition; 2) the photon's worldline is vertical; 3) the photon's proper distance is maximum (5.8 billion ly). After, this $v_{rec}$ keeps decreasing, but now the photon moves with decreasing proper distance, since $v_{rec}$ gives that $v_{tot}$ is less than zero. The photon's worlline is curving downwards.

This is a subtle example of a first-year calculus local maximum problem: $0 = \dot{D} = v_{tot}$ is a necessary condition to have a local maximum for $D$. In this eaxmple, $0 = v_{tot}$ is the condition to be on the Hubble sphere.

Also, look at the last sentence of the paragraph that contains equation (19) and (20):

"Nevertheless we can eventually receive photons that initially were receding from us because the Hubble sphere expands and overtakes the receding photons so the photons find themselves in a region with vrec < c (Section 3.3)."

 

[marcus]

Thanks G.J. this is a helpful explanation. Also the "bulge" is a good intuitive term for the midriff of the pearshaped lightcone
We should keep a link to this post handy. I or someone else will save it in the stickythread.

 

[Jorrie]

I have used Tamara's equation for comoving event horizon distance:
$$\chi_c(t) = c \int_{t}^{t_{end}} \frac{dt'}{R(t')}$$
to approximate the proper CEH and it gives a present value of 15.64 Gly.

As far as can see, in our present calculator parlance, the proper distance to the horizon will translate to:
$$D_{CEH} = \frac{1}{S} \int_{0}^{S}{\frac{dS}{H}}$$

where $H = H_0 \sqrt{\Omega_\Lambda + \Omega_m S^3 (1+S/S_{eq})}$,
as per this post.

I guess this could perhaps easily be incorporated into a calculator.

I have incorporated the CEH into CosmoLean_A22, together with a few minor upgrades. The numerical integration now has to run for scale factor 10^-7 to 10⁷, so I had to compromise on the number of steps to get an acceptable execution time (some 3 million steps). Nevertheless, the accuracy seems to be at least 3 significant digits.

Here is a "copy & paste friendly" output sample (a check-box to this effect is one of the minor upgrades):

S	a	T	T_Hub	D_now	D_then	D_hor
1.090	0.917	12.579	13.372	1.213	1.112	15.458
1.000	1.000	13.755	13.900	0.000	0.000	15.622
0.910	1.099	15.090	14.392	-1.288	-1.416	15.768
0.820	1.220	16.612	14.838	-2.604	-3.176	15.894
0.730	1.370	18.361	15.227	-3.957	-5.421	15.999
0.640	1.562	20.387	15.553	-5.343	-8.348	16.084
0.550	1.818	22.765	15.814	-6.755	-12.282	16.149
0.460	2.174	25.610	16.010	-8.187	-17.799	16.195
0.370	2.703	29.112	16.147	-9.635	-26.040	16.222
0.280	3.571	33.626	16.233	-11.092	-39.616	16.233
0.190	5.263	39.932	16.279	-12.556	-66.083	16.279
0.100	10.000	50.388	16.297	-14.022	-140.21	16.297
0.010	100.000	87.918	16.300	-15.489	-1548.8	16.300

A small amount of manual formatting may still be required. I had to remove some trailing digits from the last two lines. I'm not sure if there is a HTML table tag available in the forum editor, because that would have made it even easier to paste.

 

[marcus]

Works really well! I just now asked for a table with 110 rows, from distant past (first galaxies) to over 70 billion years into future (year 88 billion), and the table appeared instantly. I then asked for the copy-paste version and that appeared instantly. Here it is, let's see what it it looks like. this is without any aligning by hand:

S	a	T	T_Hub	D_now	D_then	D_hor
10.000	0.100	0.559	0.839	30.890	3.089	4.653
9.910	0.101	0.566	0.851	30.814	3.109	4.687
9.820	0.102	0.574	0.862	30.737	3.130	4.722
9.730	0.103	0.582	0.874	30.659	3.151	4.758
9.640	0.104	0.590	0.887	30.579	3.172	4.794
9.550	0.105	0.599	0.899	30.499	3.194	4.831
9.460	0.106	0.607	0.912	30.418	3.215	4.868
9.370	0.107	0.616	0.925	30.335	3.237	4.906
9.280	0.108	0.625	0.939	30.251	3.260	4.945
9.190	0.109	0.634	0.952	30.166	3.282	4.984
9.100	0.110	0.644	0.967	30.080	3.305	5.024
9.010	0.111	0.653	0.981	29.992	3.329	5.064
8.920	0.112	0.663	0.996	29.903	3.352	5.105
8.830	0.113	0.673	1.011	29.813	3.376	5.147
8.740	0.114	0.684	1.027	29.721	3.401	5.190
8.650	0.116	0.695	1.043	29.628	3.425	5.233
8.560	0.117	0.705	1.059	29.533	3.450	5.277
8.470	0.118	0.717	1.076	29.437	3.475	5.322
8.380	0.119	0.728	1.093	29.339	3.501	5.367
8.290	0.121	0.740	1.111	29.240	3.527	5.413
8.200	0.122	0.752	1.129	29.139	3.554	5.461
8.110	0.123	0.765	1.148	29.037	3.580	5.508
8.020	0.125	0.778	1.168	28.933	3.608	5.557
7.930	0.126	0.791	1.187	28.827	3.635	5.607
7.840	0.128	0.805	1.208	28.719	3.663	5.658
7.750	0.129	0.819	1.229	28.609	3.692	5.709
7.660	0.131	0.833	1.250	28.498	3.720	5.762
7.570	0.132	0.848	1.273	28.384	3.750	5.815
7.480	0.134	0.864	1.296	28.269	3.779	5.870
7.390	0.135	0.880	1.319	28.151	3.809	5.925
7.300	0.137	0.896	1.343	28.031	3.840	5.982
7.210	0.139	0.913	1.369	27.909	3.871	6.040
7.120	0.140	0.930	1.394	27.785	3.902	6.099
7.030	0.142	0.948	1.421	27.658	3.934	6.159
6.940	0.144	0.966	1.449	27.529	3.967	6.220
6.850	0.146	0.986	1.477	27.397	4.000	6.282
6.760	0.148	1.005	1.506	27.263	4.033	6.346
6.670	0.150	1.026	1.537	27.126	4.067	6.411
6.580	0.152	1.047	1.568	26.987	4.101	6.478
6.490	0.154	1.069	1.601	26.844	4.136	6.546
6.400	0.156	1.091	1.634	26.698	4.172	6.615
6.310	0.158	1.115	1.669	26.550	4.208	6.686
6.220	0.161	1.139	1.705	26.398	4.244	6.758
6.130	0.163	1.164	1.742	26.243	4.281	6.832
6.040	0.166	1.190	1.781	26.084	4.319	6.908
5.950	0.168	1.217	1.821	25.922	4.357	6.985
5.860	0.171	1.245	1.862	25.757	4.395	7.064
5.770	0.173	1.274	1.906	25.587	4.434	7.145
5.680	0.176	1.305	1.950	25.414	4.474	7.227
5.590	0.179	1.336	1.997	25.236	4.514	7.312
5.500	0.182	1.369	2.046	25.054	4.555	7.398
5.410	0.185	1.403	2.096	24.868	4.597	7.487
5.320	0.188	1.439	2.149	24.677	4.638	7.578
5.230	0.191	1.476	2.203	24.481	4.681	7.671
5.140	0.195	1.514	2.260	24.280	4.724	7.766
5.050	0.198	1.555	2.320	24.074	4.767	7.864
4.960	0.202	1.597	2.382	23.862	4.811	7.964
4.870	0.205	1.641	2.447	23.645	4.855	8.066
4.780	0.209	1.688	2.515	23.422	4.900	8.171
4.690	0.213	1.736	2.586	23.192	4.945	8.279
4.600	0.217	1.787	2.660	22.956	4.991	8.390
4.510	0.222	1.840	2.738	22.714	5.036	8.503
4.420	0.226	1.896	2.819	22.463	5.082	8.620
4.330	0.231	1.955	2.905	22.206	5.128	8.740
4.240	0.236	2.017	2.995	21.940	5.175	8.863
4.150	0.241	2.082	3.089	21.667	5.221	8.989
4.060	0.246	2.151	3.189	21.384	5.267	9.119
3.970	0.252	2.224	3.294	21.093	5.313	9.252
3.880	0.258	2.301	3.404	20.791	5.359	9.389
3.790	0.264	2.382	3.520	20.480	5.404	9.529
3.700	0.270	2.468	3.643	20.157	5.448	9.674
3.610	0.277	2.559	3.773	19.824	5.491	9.823
3.520	0.284	2.656	3.911	19.478	5.534	9.976
3.430	0.292	2.759	4.056	19.120	5.574	10.133
3.340	0.299	2.869	4.210	18.748	5.613	10.295
3.250	0.308	2.987	4.374	18.361	5.650	10.461
3.160	0.316	3.112	4.548	17.960	5.684	10.632
3.070	0.326	3.246	4.733	17.542	5.714	10.808
2.980	0.336	3.390	4.930	17.108	5.741	10.988
2.890	0.346	3.544	5.139	16.655	5.763	11.174
2.800	0.357	3.710	5.362	16.182	5.779	11.364
2.710	0.369	3.889	5.601	15.689	5.789	11.559
2.620	0.382	4.083	5.855	15.174	5.791	11.760
2.530	0.395	4.292	6.127	14.635	5.784	11.965
2.440	0.410	4.519	6.417	14.070	5.767	12.175
2.350	0.426	4.766	6.727	13.479	5.736	12.390
2.260	0.442	5.035	7.059	12.859	5.690	12.609
2.170	0.461	5.329	7.413	12.208	5.626	12.832
2.080	0.481	5.651	7.791	11.524	5.540	13.058
1.990	0.503	6.005	8.194	10.804	5.429	13.287
1.900	0.526	6.394	8.622	10.048	5.288	13.518
1.810	0.552	6.823	9.077	9.252	5.111	13.751
1.720	0.581	7.299	9.556	8.413	4.891	13.983
1.630	0.613	7.826	10.061	7.531	4.620	14.213
1.540	0.649	8.412	10.587	6.602	4.287	14.441
1.450	0.690	9.066	11.133	5.625	3.879	14.663
1.360	0.735	9.798	11.691	4.598	3.381	14.878
1.270	0.787	10.618	12.257	3.520	2.772	15.084
1.180	0.847	11.540	12.821	2.391	2.027	15.278
1.090	0.917	12.579	13.372	1.213	1.112	15.458
1.000	1.000	13.755	13.900	0.000	0.000	15.622
0.910	1.099	15.090	14.392	-1.288	-1.416	15.768
0.820	1.220	16.612	14.838	-2.604	-3.176	15.894
0.730	1.370	18.361	15.227	-3.957	-5.421	15.999
0.640	1.562	20.387	15.553	-5.343	-8.348	16.084
0.550	1.818	22.765	15.814	-6.755	-12.282	16.149
0.460	2.174	25.610	16.010	-8.187	-17.799	16.195
0.370	2.703	29.112	16.147	-9.635	-26.040	16.222
0.280	3.571	33.626	16.233	-11.092	-39.616	16.233
0.190	5.263	39.932	16.279	-12.556	-66.083	16.279
0.100	10.000	50.388	16.297	-14.022	-140.21	16.297
0.010	100.000	87.918	16.300	-15.489	-1548.8	16.300

Nice! Quite legible! that will be a big time saver for people who use tablulator to illustrate a point.

 

[Jorrie]

Works really well! I just now asked for a table with 110 rows, from distant past (first galaxies) to over 70 billion years into future (year 88 billion), and the table appeared instantly. I then asked for the copy-paste version and that appeared instantly. Here it is, let's see what it it looks like. this is without any aligning by hand:

S	a	T	T_Hub	D_now	D_then	D_hor
10.000	0.100	0.559	0.839	30.890	3.089	4.653
9.910	0.101	0.566	0.851	30.814	3.109	4.687
9.820	0.102	0.574	0.862	30.737	3.130	4.722
9.730	0.103	0.582	0.874	30.659	3.151	4.758
9.640	0.104	0.590	0.887	30.579	3.172	4.794
9.550	0.105	0.599	0.899	30.499	3.194	4.831
9.460	0.106	0.607	0.912	30.418	3.215	4.868
9.370	0.107	0.616	0.925	30.335	3.237	4.906
9.280	0.108	0.625	0.939	30.251	3.260	4.945
9.190	0.109	0.634	0.952	30.166	3.282	4.984
9.100	0.110	0.644	0.967	30.080	3.305	5.024
9.010	0.111	0.653	0.981	29.992	3.329	5.064
8.920	0.112	0.663	0.996	29.903	3.352	5.105
8.830	0.113	0.673	1.011	29.813	3.376	5.147
8.740	0.114	0.684	1.027	29.721	3.401	5.190
8.650	0.116	0.695	1.043	29.628	3.425	5.233
8.560	0.117	0.705	1.059	29.533	3.450	5.277
8.470	0.118	0.717	1.076	29.437	3.475	5.322
8.380	0.119	0.728	1.093	29.339	3.501	5.367
8.290	0.121	0.740	1.111	29.240	3.527	5.413
8.200	0.122	0.752	1.129	29.139	3.554	5.461
8.110	0.123	0.765	1.148	29.037	3.580	5.508
8.020	0.125	0.778	1.168	28.933	3.608	5.557
7.930	0.126	0.791	1.187	28.827	3.635	5.607
7.840	0.128	0.805	1.208	28.719	3.663	5.658
7.750	0.129	0.819	1.229	28.609	3.692	5.709
7.660	0.131	0.833	1.250	28.498	3.720	5.762
7.570	0.132	0.848	1.273	28.384	3.750	5.815
7.480	0.134	0.864	1.296	28.269	3.779	5.870
7.390	0.135	0.880	1.319	28.151	3.809	5.925
7.300	0.137	0.896	1.343	28.031	3.840	5.982
7.210	0.139	0.913	1.369	27.909	3.871	6.040
7.120	0.140	0.930	1.394	27.785	3.902	6.099
7.030	0.142	0.948	1.421	27.658	3.934	6.159
6.940	0.144	0.966	1.449	27.529	3.967	6.220
6.850	0.146	0.986	1.477	27.397	4.000	6.282
6.760	0.148	1.005	1.506	27.263	4.033	6.346
6.670	0.150	1.026	1.537	27.126	4.067	6.411
6.580	0.152	1.047	1.568	26.987	4.101	6.478
6.490	0.154	1.069	1.601	26.844	4.136	6.546
6.400	0.156	1.091	1.634	26.698	4.172	6.615
6.310	0.158	1.115	1.669	26.550	4.208	6.686
6.220	0.161	1.139	1.705	26.398	4.244	6.758
6.130	0.163	1.164	1.742	26.243	4.281	6.832
6.040	0.166	1.190	1.781	26.084	4.319	6.908
5.950	0.168	1.217	1.821	25.922	4.357	6.985
5.860	0.171	1.245	1.862	25.757	4.395	7.064
5.770	0.173	1.274	1.906	25.587	4.434	7.145
5.680	0.176	1.305	1.950	25.414	4.474	7.227
5.590	0.179	1.336	1.997	25.236	4.514	7.312
5.500	0.182	1.369	2.046	25.054	4.555	7.398
5.410	0.185	1.403	2.096	24.868	4.597	7.487
5.320	0.188	1.439	2.149	24.677	4.638	7.578
5.230	0.191	1.476	2.203	24.481	4.681	7.671
5.140	0.195	1.514	2.260	24.280	4.724	7.766
5.050	0.198	1.555	2.320	24.074	4.767	7.864
4.960	0.202	1.597	2.382	23.862	4.811	7.964
4.870	0.205	1.641	2.447	23.645	4.855	8.066
4.780	0.209	1.688	2.515	23.422	4.900	8.171
4.690	0.213	1.736	2.586	23.192	4.945	8.279
4.600	0.217	1.787	2.660	22.956	4.991	8.390
4.510	0.222	1.840	2.738	22.714	5.036	8.503
4.420	0.226	1.896	2.819	22.463	5.082	8.620
4.330	0.231	1.955	2.905	22.206	5.128	8.740
4.240	0.236	2.017	2.995	21.940	5.175	8.863
4.150	0.241	2.082	3.089	21.667	5.221	8.989
4.060	0.246	2.151	3.189	21.384	5.267	9.119
3.970	0.252	2.224	3.294	21.093	5.313	9.252
3.880	0.258	2.301	3.404	20.791	5.359	9.389
3.790	0.264	2.382	3.520	20.480	5.404	9.529
3.700	0.270	2.468	3.643	20.157	5.448	9.674
3.610	0.277	2.559	3.773	19.824	5.491	9.823
3.520	0.284	2.656	3.911	19.478	5.534	9.976
3.430	0.292	2.759	4.056	19.120	5.574	10.133
3.340	0.299	2.869	4.210	18.748	5.613	10.295
3.250	0.308	2.987	4.374	18.361	5.650	10.461
3.160	0.316	3.112	4.548	17.960	5.684	10.632
3.070	0.326	3.246	4.733	17.542	5.714	10.808
2.980	0.336	3.390	4.930	17.108	5.741	10.988
2.890	0.346	3.544	5.139	16.655	5.763	11.174
2.800	0.357	3.710	5.362	16.182	5.779	11.364
2.710	0.369	3.889	5.601	15.689	5.789	11.559
2.620	0.382	4.083	5.855	15.174	5.791	11.760
2.530	0.395	4.292	6.127	14.635	5.784	11.965
2.440	0.410	4.519	6.417	14.070	5.767	12.175
2.350	0.426	4.766	6.727	13.479	5.736	12.390
2.260	0.442	5.035	7.059	12.859	5.690	12.609
2.170	0.461	5.329	7.413	12.208	5.626	12.832
2.080	0.481	5.651	7.791	11.524	5.540	13.058
1.990	0.503	6.005	8.194	10.804	5.429	13.287
1.900	0.526	6.394	8.622	10.048	5.288	13.518
1.810	0.552	6.823	9.077	9.252	5.111	13.751
1.720	0.581	7.299	9.556	8.413	4.891	13.983
1.630	0.613	7.826	10.061	7.531	4.620	14.213
1.540	0.649	8.412	10.587	6.602	4.287	14.441
1.450	0.690	9.066	11.133	5.625	3.879	14.663
1.360	0.735	9.798	11.691	4.598	3.381	14.878
1.270	0.787	10.618	12.257	3.520	2.772	15.084
1.180	0.847	11.540	12.821	2.391	2.027	15.278
1.090	0.917	12.579	13.372	1.213	1.112	15.458
1.000	1.000	13.755	13.900	0.000	0.000	15.622
0.910	1.099	15.090	14.392	-1.288	-1.416	15.768
0.820	1.220	16.612	14.838	-2.604	-3.176	15.894
0.730	1.370	18.361	15.227	-3.957	-5.421	15.999
0.640	1.562	20.387	15.553	-5.343	-8.348	16.084
0.550	1.818	22.765	15.814	-6.755	-12.282	16.149
0.460	2.174	25.610	16.010	-8.187	-17.799	16.195
0.370	2.703	29.112	16.147	-9.635	-26.040	16.222
0.280	3.571	33.626	16.233	-11.092	-39.616	16.233
0.190	5.263	39.932	16.279	-12.556	-66.083	16.279
0.100	10.000	50.388	16.297	-14.022	-140.21	16.297
0.010	100.000	87.918	16.300	-15.489	-1548.8	16.300

For better visualization, here is a spreadsheet graph of your table over a useful range:

The visual relationships between T, T_Hubble and D_horizon are particularly interesting. I guess a lot can be written about it.

PS: I have re-uploaded CosmoLean_A22, fixing some practical problems with the input-alert pop-ups; I have removed them completely. In their place I am changing the color of the 'range-text' to red when the values are out of range. This is less annoying and compatible with pop-up blockers. I have not changed designation, so no need to change your sig.

I have also used the opportunity to add one useful piece of info: the Hubble constant corresponding the Ynow input, so that we can get a feel for what our inputs are doing to the conventional values.

I am contemplating the addition of two other corresponding conventional values, i.e. $\Omega_\Lambda$ and $\Omega_m$. What do you think?