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## Cosmo calculators with tabular output

 Quote by marcus So the rule should be to have at least 3 steps? Or 5 steps? When venturing into high S territory please make the step size less than some fraction of the range, like 1/3 or 1/5?
It also depends on how high is high S. It looks like problems start to occur when a single step represents some 0.1% of S. The problem may however be solved in the next update, e.g. by making it small enough or by catching and preventing it from causing trouble.

I am working on flexible rounding of output column data; it's working, but a few issues still prevent it from being released.
 Recognitions: Gold Member Science Advisor Since we just turned a page, I will bring forward the sample table output from post#12 earlier and also copy some relevant comment. At this point we are mostly talking about how to use the new tabular-output calculator. It can be used basically as a one-shot if that is all you want but there are things you can see from a table. Also it's nice having the input be a range of (reciprocal) scale. ===quote Jorrie; 4059998=== ... I also prefer the stretch or scale factor over time for more than one reason. Firstly, it is relatively easy to visualize the matter-radiation equality epoch at some 1/3350 th of the present scale, but how easy is it to visualize 50 thousand years on a scale of 14 billion years? Secondly, cosmic models run more efficiently with scale factor as independent variable; we know the limits in advance, being 'a' from near 0 to 1, with the upper limit model independent. Time runs from near zero to some unknown time today, which is model dependent. ===endquote=== ===marcus;4060517=== This is a clear statement of motivation and could be included in an online "user's booklet" for the CosmoLean if there were one. Another thing that would be nice in such a booklet would be this figure: http://ned.ipac.caltech.edu/level5/M...es/figure1.jpg Because when you make a table like the one I just posted some of the columns correspond to curves in the figure. For example look at the middle strip of the figure. the Dnow column corresponds to the LIGHTCONE curve. That spreads farther and farther out as you go back further to smaller scalefactor. that is because Dnow is the same as comoving distance, and you can see that scalefactor is the measure plotted along the righthand side of the strip. As a check, looking back at post #12 you see from the table that by the time you get down to scale 0.1 the comove distance of the lightcone should be around 31 Gly. So let's look at Lineweaver's figure and see. Yes. It checks. Lineweaver's 2003 parameters are not exactly the same as the 2010 ones so the plot does not exactly agree but it's pretty close. You can see the agreement even better on the lower strip, which also has comoving distance. but has the scalefactor marks more spread out. It is easier to find scale=0.1 on the righthand edge of the strip (i.e. S=10) Also the Dthen column of the table in post #12 should correspond to the lightcone in the TOP strip because in that one the distance coordinate is PROPER distance. The lightcone should bulge out to 5.8 Gly at around scale 0.375 (S=2.666) and then should be back to 3 Gly by the time it gets to scale 0.1 (S=10). So let's check. Well the figure is a bit cramped and smudgy but it looks about right. there is only a tick-mark at proper distance 10 Gly, so you have to judge by eye where 5.8 is. ===endquote=== I will go fetch a copy of that sample output, so readers can see what we're talking about.
 Recognitions: Gold Member Science Advisor I'll bring forward post#12 of previous page ===quote marcus;4060562=== So far my favorite output table is where you put start S = 1 (i.e. present) end S = 10 (i.e. first galaxies forming, distances 1/10 of today size) step = 0.33333 (five digits is enough to get effective step of 1/3) then what you get is this: Code: S=1/a scalefactor a time(Gy) Hubbletime(Gy) Dnow(Gly) Dthen(Gly) 10.00 0.100000 0.558619 0.839348 30.904551 3.090455 9.67 0.103448 0.587799 0.883047 30.617708 3.167349 9.33 0.107143 0.619654 0.930686 30.315192 3.248056 9.00 0.111111 0.654446 0.982733 29.996387 3.332932 8.67 0.115385 0.692615 1.039801 29.659359 3.422234 8.33 0.120000 0.734549 1.102548 29.303064 3.516368 8.00 0.125000 0.780996 1.171897 28.923900 3.615487 7.67 0.130435 0.832503 1.248777 28.520615 3.720080 7.33 0.136364 0.889918 1.334397 28.090224 3.830485 7.00 0.142857 0.954152 1.430165 27.630118 3.947160 6.67 0.150000 1.026561 1.537915 27.135608 4.070341 6.33 0.157895 1.108514 1.659755 26.603238 4.200511 6.00 0.166667 1.201987 1.798433 26.027216 4.337869 5.67 0.176471 1.309229 1.957280 25.402101 4.482723 5.33 0.187500 1.433317 2.140615 24.720163 4.635030 5.00 0.200000 1.578263 2.353993 23.971943 4.794388 4.67 0.214286 1.749255 2.604580 23.146333 4.959928 4.33 0.230769 1.953045 2.901717 22.230355 5.130081 4.00 0.250000 2.199343 3.258071 21.205492 5.301372 3.67 0.272727 2.501266 3.690535 20.049940 5.468165 3.33 0.300000 2.877818 4.222240 18.734447 5.620333 3.00 0.333333 3.356917 4.884836 17.220673 5.740223 2.67 0.375000 3.980585 5.721191 15.458441 5.796914 2.33 0.428571 4.814342 6.787256 13.381146 5.734775 2.00 0.500000 5.964059 8.147995 10.900901 5.450448 1.67 0.600000 7.604379 9.852421 7.910657 4.746392 1.33 0.750000 10.030831 11.858689 4.298519 3.223887 1.00 0.999999 13.754769 13.899959 0.000026 0.000026 Now here's a neat thing: we can read off the COMOVING HUBBLE RADIUS at various past epochs from this. YOU JUST HAVE TO MULTIPLY S TIMES THE HUBBLETIME that corresponds to that stretch S!!! I like this feature. the output is lean but also rich in possibilities. For example for S=10 the Hubbletime is 0.84 Gy, so you get 8.4 Gly and for S = 1.67 the Hubbletime is 9.85 Gy, so by multiplying you get 16.4 Gly. Now to check that we can go to Lineweaver's figure 1 because he plots curves for things like the lightcone and the Hubble radius in comoving distance. And it checks! You see that 1/1.67 = 0.6 and look at the bottom strip of the figure http://ned.ipac.caltech.edu/level5/M...es/figure1.jpg at the level marked scale = 0.6, and behold, the comoving distance of the Hubble radius is about 16 Gly. And at the level marked scale = 0.1, the Hubble radius should be about 8.4 according to the calculator's table, and so it is. ===endquote=== So to summarize, what we're seeing is that you can read stuff off the table that corresponds to the curves in Lineweaver's Figure 1 http://ned.ipac.caltech.edu/level5/M...es/figure1.jpg namely the lightcone in proper distance the lightcone in comoving (now) distance the Hubble radius in proper distance (just interpret the Hubbletime Gy as Gly) the Hubble radius in comoving distance (just multiply by S) These things are a cinch to read directly off the table. Other things you can get from the table are "recession speeds" (now or then)as multiples of the speed of light. Just divide the then distance by the then Hubbletime, or divide the now distance by the now Hubbletime. They should not be thought of as speeds of anything traveling in the usual sense, but as the speeds distances are growing. Further things you can get from the table are the angles which something of a given size makes in the sky (which will be found using the table, from its wavelength stretch).

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 Quote by Jorrie ... It looks like problems start to occur when a single step represents some 0.1% of S. The problem may however be solved in the next update, e.g. by making it small enough or by catching and preventing it from causing trouble. I am working on flexible rounding of output column data; it's working, but a few issues still prevent it from being released.
The 'single-step problem' has been solved in CosmoLean_A17 and the 'few issues' with the flexible rounding of column data are gone as well, or so I hope. Please try it out and report any anomalies.

The most important differences are:
1. The info-popups have been mostly reworded and include comments as received in PMs.
2. The stretch range inputs arranged to be more consistent with the output table, from highest to lowest stretch.
3. Some 'logic' built into the input processor so that 'one-shot' outputs are intuitively achieved by either making s_step zero, which gives output for s_upper only; or by making s_upper and s_lower equal to each other, irrespective of s_step.
4. The number of decimals (rounding) of column data are adjustable individually. Becomes active on clicking Calculate and will remain so until changed again, reset clicked, or the page is refreshed.
5. Overall accuracy has been improved by resolving some coding issues. It now seems to work accurately up to s =10 000.
6. Some input validation and protection against crash of program are included. More to be considered.
7. On the drawing board: "into the future" (s < 1).
 Marcus / Jorrie, Would you mind posting the link again please? I'm afraid that the mobile version of PF does not include your signatures so I'm having difficulty finding it. Regards, Noel.

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 Quote by Lino Marcus / Jorrie, Would you mind posting the link again please? I'm afraid that the mobile version of PF does not include your signatures so I'm having difficulty finding it. Regards, Noel.
Did you mean the link to the calculator? If so, it is in my prior post, labelled: CosmoLean_A17, but I copied it here. The one in Marcus's sig may still be the old release.
 Thanks Jorrie. Much appreciated. Regards, Noel.
 Recognitions: Gold Member Science Advisor The new version is a pleasure to use. We should accumulate a bit of "user manual" type information like that in your post#21, three or four posts back. Putting step=0 makes it very simple to use as a one-shot. It's mostly self-explanatory how to use it, so not very much by way of "user manual" seems necessary. but at least the hint about setting step to zero. The feature of deciding on how many decimal places to show is quite nice. It rounds off for you. I like seeing 3-place precision but knowing I'm riding on 6-place (like a new set of tires on the car, you just feel better.) Visually clean, sufficient but just what's essential.

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 Quote by marcus The new version is a pleasure to use. We should accumulate a bit of "user manual" type information like that in your post#21, three or four posts back.
Good idea. Will add a on-page popup in next update with some general tips.

BTW, work-shopping the 'back to the future' options on a spreadsheet, it struck me that things like D_now and D_then are then ill defined. For the past is is easy; we think in terms of a source observed at (say) stretch 2 and we ask the questions: how far was it at time of emission and how far is it now?

The equivalents for the future are more obscure. Since we cannot observe anything from the future, we may have to think in terms of emitting some signal now and answer the question: How far must an observer be from us to receive that signal at a stretch of (say) 2, both now and then?

Or is there a better way?

PS: Lean gives for s=2, D_now=11.1 Gly and D_then half of this. For a future s=1/2, I get via spreadsheet: D_now=7.4 and D_then twice this.

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 Quote by Jorrie The 'single-step problem' has been solved in CosmoLean_A17 and the 'few issues' with the flexible rounding of column data are gone as well, or so I hope. Please try it out and report any anomalies. ...
The latest version (as in Marcus's signature) is CosmoLean_A20, which adds an 'Introduction' button with some hints for usage. It is supposed to be fairly stable now and it is perhaps time to give an idea of the underlying formulas and conventions. It follows the development of the 13.9/16.3 simplified model proposed by Marcus, but with inclusion of the early stage radiation energy density.

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

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

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

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

For perfect flatness, it can be expressed as

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

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

From H, the following calculator outputs are readily available:

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

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

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

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

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

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

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

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

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 Quote by Jorrie ... In principle, the equations can be used for projecting into the future as well. This has been "secretly" sneaked into version A20. If you want to try it out, enter 1 into S_upper and 0.1 into both S_lower and S_Step. Note the time going to some 50 Gy, T_Hubble to around 16.3 Gy and the distances to negative values. As Marcus has pointed out before, D_now for this scenario is the present distance to a target that will receive our signals with a wavelength stretch S at future time T(a). D_then means the proper distance of the target when they eventually receive our signal, obviously 1/S times farther. This 'trial feature' can go down to S = 0.01 in steps of 0.01, but not lower at this time.
Hand-in-hand with the 'future option' goes the cosmic event horizon. It has been included in CosmoLean_A22.

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

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

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

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

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

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

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

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 Quote by Jorrie Hand-in-hand with the 'future option' goes the cosmic event horizon. It has been included in CosmoLean_A22.
Following some tests and hints from others, some changes have been made to the user interface and it is now at version Cosmolean_A25.

The troublesome "range check pop-ups" of the input boxes have been replaced by a feature that simply changes the color of the "range text" to red when a problem is detected. Out of range values and non-numerical inputs are actually accepted and you can calculated with them if you wish. AFAIK, the calculator do not crash, but may throw out funny values.

At the same time the conventional values of Ho, Omega_Lambda and Omega_matter is calculated and displayed so that the influence of your (changed) input becomes visible to you. Also, 'down-step' values S_step <= 0 now have the special meaning of setting the number of steps between S_upper and S_lower to the rounded-abs of the value given. A 'one-shot' (single row) is hence still produced by S_step = zero.

Here is what it the input section now looks like.

The outputs with 100 steps:

Code:
S	a	T	T_Hub	D_now	D_then	D_hor
10.00	0.100	0.559	0.839	30.890	3.089	4.653
9.90	0.101	0.567	0.852	30.805	3.112	4.691
9.80	0.102	0.576	0.865	30.720	3.135	4.730
9.70	0.103	0.585	0.878	30.633	3.158	4.770
9.60	0.104	0.594	0.892	30.544	3.182	4.810
9.50	0.105	0.603	0.906	30.454	3.206	4.851
9.40	0.106	0.613	0.921	30.363	3.230	4.893
9.30	0.108	0.623	0.936	30.270	3.255	4.936
9.20	0.109	0.633	0.951	30.176	3.280	4.979
9.10	0.110	0.643	0.966	30.080	3.305	5.023
9.00	0.111	0.654	0.983	29.983	3.331	5.068
8.90	0.112	0.665	0.999	29.884	3.357	5.114
8.80	0.114	0.677	1.016	29.783	3.384	5.161
8.70	0.115	0.688	1.034	29.681	3.411	5.208
8.60	0.116	0.700	1.052	29.577	3.439	5.257
8.50	0.118	0.713	1.070	29.471	3.467	5.306
8.40	0.119	0.726	1.089	29.363	3.495	5.356
8.30	0.120	0.739	1.109	29.253	3.524	5.407
8.20	0.122	0.752	1.129	29.142	3.553	5.460
8.10	0.123	0.766	1.150	29.028	3.583	5.513
8.00	0.125	0.781	1.171	28.912	3.613	5.567
7.90	0.127	0.795	1.194	28.794	3.644	5.623
7.80	0.128	0.811	1.217	28.673	3.675	5.679
7.70	0.130	0.827	1.240	28.550	3.707	5.737
7.60	0.132	0.843	1.265	28.425	3.739	5.796
7.50	0.133	0.860	1.290	28.298	3.772	5.856
7.40	0.135	0.877	1.316	28.168	3.805	5.917
7.30	0.137	0.895	1.343	28.035	3.839	5.980
7.20	0.139	0.914	1.371	27.899	3.873	6.044
7.10	0.141	0.933	1.399	27.761	3.908	6.110
7.00	0.143	0.953	1.429	27.620	3.944	6.177
6.90	0.145	0.974	1.460	27.475	3.980	6.245
6.80	0.147	0.996	1.492	27.328	4.017	6.315
6.70	0.149	1.018	1.525	27.177	4.054	6.387
6.60	0.151	1.041	1.560	27.023	4.092	6.460
6.50	0.154	1.065	1.596	26.866	4.131	6.535
6.40	0.156	1.090	1.633	26.704	4.170	6.612
6.30	0.159	1.116	1.671	26.539	4.210	6.691
6.20	0.161	1.143	1.711	26.370	4.251	6.771
6.10	0.164	1.171	1.753	26.197	4.292	6.854
6.00	0.167	1.201	1.797	26.020	4.334	6.938
5.90	0.169	1.231	1.842	25.838	4.376	7.025
5.80	0.172	1.263	1.889	25.652	4.420	7.114
5.70	0.175	1.296	1.938	25.461	4.463	7.205
5.60	0.178	1.331	1.990	25.265	4.508	7.298
5.50	0.182	1.367	2.043	25.063	4.553	7.394
5.40	0.185	1.405	2.099	24.856	4.599	7.492
5.30	0.189	1.445	2.158	24.644	4.646	7.593
5.20	0.192	1.486	2.219	24.425	4.693	7.697
5.10	0.196	1.530	2.283	24.200	4.741	7.804
5.00	0.200	1.576	2.351	23.969	4.789	7.913
4.91	0.204	1.624	2.421	23.731	4.838	8.026
4.81	0.208	1.675	2.495	23.485	4.887	8.142
4.71	0.213	1.728	2.573	23.232	4.937	8.261
4.61	0.217	1.784	2.655	22.971	4.988	8.383
4.51	0.222	1.843	2.742	22.701	5.039	8.509
4.41	0.227	1.906	2.833	22.423	5.090	8.639
4.31	0.232	1.972	2.929	22.135	5.141	8.773
4.21	0.238	2.042	3.030	21.837	5.192	8.910
4.11	0.244	2.116	3.137	21.529	5.244	9.052
4.01	0.250	2.194	3.251	21.210	5.295	9.198
3.91	0.256	2.278	3.371	20.880	5.345	9.349
3.81	0.263	2.367	3.499	20.537	5.396	9.504
3.71	0.270	2.462	3.634	20.180	5.445	9.664
3.61	0.277	2.563	3.779	19.810	5.493	9.829
3.51	0.285	2.671	3.932	19.425	5.540	9.999
3.41	0.294	2.787	4.095	19.024	5.584	10.175
3.31	0.302	2.912	4.270	18.606	5.627	10.356
3.21	0.312	3.046	4.456	18.171	5.666	10.542
3.11	0.322	3.190	4.656	17.716	5.702	10.735
3.01	0.333	3.345	4.869	17.240	5.733	10.933
2.91	0.344	3.514	5.098	16.742	5.759	11.138
2.81	0.356	3.696	5.344	16.221	5.778	11.349
2.71	0.369	3.895	5.608	15.674	5.789	11.565
2.61	0.384	4.111	5.892	15.100	5.791	11.788
2.51	0.399	4.347	6.198	14.496	5.781	12.017
2.41	0.415	4.606	6.527	13.861	5.757	12.252
2.31	0.433	4.890	6.881	13.191	5.716	12.492
2.21	0.453	5.203	7.262	12.485	5.655	12.737
2.11	0.474	5.548	7.671	11.739	5.569	12.987
2.01	0.498	5.931	8.111	10.951	5.454	13.241
1.91	0.524	6.357	8.583	10.118	5.302	13.497
1.81	0.553	6.832	9.086	9.235	5.107	13.755
1.71	0.585	7.364	9.621	8.301	4.859	14.013
1.61	0.622	7.961	10.185	7.312	4.546	14.268
1.51	0.663	8.633	10.776	6.265	4.153	14.519
1.41	0.710	9.392	11.388	5.158	3.662	14.763
1.31	0.764	10.253	12.014	3.990	3.048	14.997
1.21	0.827	11.233	12.641	2.758	2.282	15.218
1.11	0.902	12.350	13.258	1.464	1.320	15.422
1.01	0.991	13.630	13.849	0.110	0.109	15.606
0.91	1.100	15.104	14.397	-1.301	-1.431	15.769
0.81	1.236	16.809	14.887	-2.765	-3.416	15.908
0.71	1.410	18.800	15.308	-4.273	-6.025	16.021
0.61	1.641	21.152	15.649	-5.820	-9.551	16.109
0.51	1.963	23.979	15.910	-7.397	-14.519	16.172
0.41	2.441	27.473	16.094	-8.997	-21.964	16.212
0.31	3.229	31.991	16.210	-10.611	-34.261	16.230
0.21	4.766	38.318	16.272	-12.233	-58.310	16.272
0.11	9.099	48.849	16.296	-13.860	-126.118 16.296
0.01	100.000	87.918	16.300	-15.489	-1548.864 16.300
Here are nice graphs of most of those columns:

Anyone with means of checking the validity of the outputs? I have a suspicion that D_now is not correct for the future (S < 1), because I think it is supposed to asymptotically approach the S = 0 line, while it appears to be heading for an intercept. The current calculator does not work accurately for S < 0.01, which may actually be the cause of the apparent intercept. Looking into it.

Edit:
The "y-intercept" of the green curve is in fact just an artefact of this thread's definition of D_now for the future: the (negative) distance to an observer that will receive our present signals with a stretch 1/S, i.e. with redshift 1/S + 1. The y-intercept represents the cosmic even horizon (16.3 Gy), where redshift (and time to reach) tends to infinity. Negative S does not have a physical meaning, or does it? One can mathematically extend the curve to the negative domain, but I have no idea what it may mean.
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Hi Jorrie, I just saw your edit. I think you are right that the curve physically stops at 16.3 where the time for the signal to reach target goes to infinity. If it takes an infinite time for a our signal to reach a galaxy at 16.3 Gly that clearly says it is the limit. I like the clarity.

Can't think of any physical meaning of negative stretch, or negative scale factor.

To me it looks like the calculator does what it has to do, what it should do. reach the axis (where time=) exactly at the right place. It's a really satisfying gadget, you must be be having some proud papa moments these days.

(Or so it seems to me---as a non-expert interested in the subject.)

btw I like the "down-step" feature! It lets me get the size table I want without having to calculate what the step size should be to achieve that. And when I change the upper and lower limits of the table, it stays the desired size. Good (though unconventional) use of the minus sign

EDIT: Hi Jorrie, just saw your next post which wakes me up to the fact that I should have been saying 15.6 here instead of 16.3. The y-intercept of the D_now curve should give the present value of the cosmic event horizon (which is around 15.6 Gly) not the future value.
 Quote by Jorrie Here are nice graphs of most of those columns: Edit: The "y-intercept" of the green curve is in fact just an artefact of this thread's definition of D_now for the future: the (negative) distance to an observer that will receive our present signals with a stretch 1/S, i.e. with redshift 1/S + 1. The y-intercept represents the cosmic even horizon (16.3 Gy), where redshift (and time to reach) tends to infinity. Negative S does not have a physical meaning, or does it? One can mathematically extend the curve to the negative domain, but I have no idea what it may mean.

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 Quote by Jorrie ... I have a suspicion that D_now is not correct for the future (S < 1), because I think it is supposed to asymptotically approach the S = 0 line, while it appears to be heading for an intercept. ...
I was wrong; it is actually the future D_then value that diverges to negative infinity, as is clear from this chart.

One can also see the position of the 'equator' of the usual 'teardrop' (or 'onion') shape, where T_Hub and D_then cross over (S = 2.64). This is the maximum value of D_then for all of time (given the standard model and values).

 Quote by Jorrie ... Edit: The "y-intercept" of the green curve is in fact just an artefact of this thread's definition of D_now for the future: the (negative) distance to an observer that will receive our present signals with a stretch 1/S, i.e. with redshift 1/S + 1. The y-intercept represents the cosmic even horizon (16.3 Gy), where redshift (and time to reach) tends to infinity. ...
Wrong again; must have been weekend laziness...

The "y-intercept" of the green curve represents (the negative of) the distance to our or present cosmic event horizon (CEH), at 15.6 Gly. An observer presently at that proper distance will never receive our present signals (and neither will we receive theirs). If accelerated expansion continues as we expect, our future CEH will only reach 16.3 Gly by around 74 Gy from now.
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 Recognitions: Gold Member Science Advisor That's an especially nice chart with the 5 curves (time=black, horizon=sky blue, ...etc). I like being able to spot the equatorial bulge on the onionshape lightcone by where the red and purple curves cross, at S=2.64. Your post alerts me to my having misspoke in post#31, the current CEH being 15.6, I should have been saying that instead of the longterm CEH value of 16.3. The presentday D_now curve should have a y-intercept at the presentday CEH, so at or around -15.6. Which (allowing for the limitations of finite accuracy) it does seem to do!

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Since there may be a change in the generally accepted values for H0 and Omegam coming (http://arxiv.org/abs/1208.3281%22), and it may change values for i.a. Cosmic time (age) and Lookback time, I have included Lookback time in the list of compact equations that was listed before.
 Quote by Jorrie Given present Hubble time $Y_{now}$, long term Hubble time $Y_{inf}$ and the redshift for radiation/matter equality $z_{eq}$. Since the factor $z + 1$ occurs so often, an extra parameter $S = z + 1 = 1/a$ is defined, making the equations neater. $$\Omega_\Lambda = \left(\frac{Y_{now}}{Y_{inf}}\right)^2, \ \ \, \Omega_r = \frac{1-\Omega_\Lambda}{1+S_{eq}}, \ \ \, \Omega_m = S_{eq}\Omega_r$$ Hubble parameter $$H = H_0 \sqrt{\Omega_\Lambda + \Omega_m S^3 (1+S/S_{eq})}$$ Hubble time, Cosmic time , Lookback time Tlook added $$Y = 1/H, \ \ \, T = \int_{S}^{\infty}{\frac{dS}{S H}}, \ \ \, T_{look} = \int_{1}^{S}{\frac{dS}{S H}}$$ Proper distance 'now', 'then' and cosmic event horizon $$D_{now} = \int_{1}^{S}{\frac{dS}{H}}, \ \ \, D_{then} = \frac{D_{now}}{S}, \ \ \, D_{CEH} = \frac{1}{S} \int_{0}^{S}{\frac{dS}{H}}$$ This essentially means integration for S from zero to infinity, but practically it has been limited to $10^{-7} < S < 10^{7}$ with quasi-logarithmic step sizes, e.g. a small % increase between integration steps.
Using the quoted H0 = 74.3 km s-1 Mpc-1 and Omegam = 0.278, a rerun of the above numerical integrations gives:

Hubble times: Ynow = 13.3 Gyr, Yinf = 15.5 Gyr, Tnow = 12.96 Gyr and the lookback time to the current most distant galaxy Tz=9.6 = 12.54 Gyr.

Since the change in Omegam was small, the times essentially changed by the ratio H0(old)/H0(new), but this will not hold if Omegam changes significantly, or the deviation from spatial flatness is significant.