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Effort to get us all on the same page (balloon analogy) |
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| Aug19-12, 10:25 PM | #409 |
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Effort to get us all on the same page (balloon analogy)
I woke up (yes, it's rise and shine time here already), with the subconscious telling me the following: convert the new inputs to the old inputs behind the scenes and perform the proper Friedman calculation à la the old calculator. Then pump out a simplified set of results and give your approximation equations in info popups, with some caveats. How would this sit with you? PS: It's also a lot less work... ;-) |
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| Aug19-12, 10:39 PM | #410 |
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I imagine I would be delighted.
The main requirement is that the project make sense to you (the calculator builder) and that you be satisfied with the results. I was intrigued by the idea of a calculator (possibly quite simple) that would not outwardly involve H and Omega_Lambda. It would let you control the current growth rate and the future growth rate by entering two Hubbletimes, instead. Anything that does this, however it does it, seems like an interesting pedagogical tool. No outward reference to "km/s" and "dark energy". Instead: a current percentage growth rate and a future asymptotic one. I think Hubbletimes are probably the easiest handles to use, to specify current and future growth rates. So I immediately think of being able to input, say, 13.9 Gy and 16.3 Gy. and maybe the presentday ratio of matter to radiation, and that's it. After that I can convert any redshift to several outputs, or an expansion age. But you are the one who has worked on cosmology calculators so you will have your own criteria and ways to reckon how well things will communicate to the user. You're the one with experience, so you be the judge. As I recall I found that 13.9 and 16.3 corresponded to something like 70.35 km/s per Mpc and .7272 for Omega_Lambda. Then Omega matter was 0.2727188 And Omega radiation was 0.0000812 So they added up to 0.2728, giving flatness. That means the matter/radiation ratio was 3359*. (Which is why you need to go back to a redshift of around 3350 or 3360 in order for them to be on par with each other.) So if I was using your calculator I would like to be able to input 13.9 billion and 16.3 billion and a number like 3360. Then the calculator would secretly change (13.9, 16.3, 3360) inputs for the existing program and proceed from there. I'm getting curious to see how this takes shape! It is like constructing a new "front end" for something you already have that runs well. Let me check that my memory was accurate about that 70.35... I paste this into google: 1/(13.9 billion years) in km/s per Mpc Yes! it immediately comes back with "70.3463274 (km/s) per Mpc" Since it is internal, perhaps better to use 70.34633 or something like that. And google also tells me that 13.9^2/16.3^2 = 0.727200873 *calculated from 2727188/812=3358.6... |
| Aug20-12, 01:18 AM | #411 |
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Updated expansion table based on treating now and future Hubbletimes 13.9 and 16.3 billion years as exact. In the following calculations, 0.7272 is the square of their ratio, and a time unit (d) is used which is a tenth of a billion years. Age at present 137.574 d. The model has to be considered a "toy" because the simplified equations give only rough approximation at times before 1 d. I'm trying out an additional column for the scalefactor at = 1/(1+zt) which shows the growth of a generic distance.
Google codes used are: at = (1/.7272-1))^(1/3)/((tanh((t+.001)*1.5/163))^-2 - 1)^(1/3) zt = ((tanh((t+.001)*1.5/163))^-2 - 1)/(1/.7272-1))^(1/3) - 1 Yt = 163 tanh((t+.001)*1.5/163) To calculate with a code, paste blue expression into google, replace t by an expansion age, and press =. EXPANSION HISTORY, 139/163 MODEL. Code:
Age Redshift Hubble time Scale factor t (d) zt Yt (d) at 0.0030 1252 0.00600 0.0008 0.0039 1093 0.00735 0.0009 1 30.553 1.501 0.032 2 18.883 3.001 0.050 3 14.175 4.500 0.066 4 11.526 5.999 0.080 5 9.794 7.496 0.093 6 8.558 8.992 0.105 7 7.624 10.487 0.116 8 6.888 11.980 0.127 9 6.291 13.471 0.137 10 5.796 14.959 0.147 20 3.269 29.667 0.234 30 2.243 43.892 0.308 40 1.659 57.431 0.376 50 1.273 70.121 0.440 60 0.992 81.848 0.502 70 0.776 92.542 0.563 80 0.603 102.176 0.624 90 0.459 110.762 0.685 100 0.337 118.341 0.748 110 0.232 124.973 0.812 120 0.139 130.732 0.878 130 0.057 135.703 0.946 131 0.049 136.159 0.953 132 0.041 136.609 0.960 133 0.034 137.052 0.967 134 0.026 137.488 0.974 135 0.019 137.918 0.981 136 0.012 138.341 0.989 137 0.004 138.757 0.996 137.574 0.000 138.993 1.000 zt = ((tanh(t*2/163)^-2 - 1)/(1/.7272-1))^(1/3) - 1 Yt = 163 tanh(t*2/163) The same caveat applies. Only a rough approximation to early universe behavior. Some notes on the table: z : fractional amount distances and wavelengths have increased while light was in transit. Arriving wavelength is 1+z times original. H=1/Y: Hubble expansion rate. Distances between stationary observers grow at this fractional rate--a certain fraction or percentage of their length per unit time. H(per d) : fractional increase per convenient unit of time d = 108 years. Y=1/H: Hubble time. 1% of the current Hubble time is how long it would take for distances to increase by 1%, growing at current rate. At present, Y is 139 d = 13.9 billion years. Hubble time is proportional to the Hubble radius = c/H: distances smaller than this grow slower than the speed of light. At present, the Hubble radius is 13.9 billion ly (proper distance) The Hubble law describes the expansion of distances between observers at rest with respect to the background of ancient light and the expansion process itself: Observers who see the ancient light and the expansion process approximately the same in all directions, e.g. no Doppler hotspots. The field of an observer's view can be thought of as pear-shape because distances were shorter back then. Here is a picture of an Anjou pear. http://carrotsareorange.com/wp-conte...pear-anjou.jpg Here is Lineweaver's spacetime diagram: http://ned.ipac.caltech.edu/level5/M...es/figure1.jpg The upperstory figure, with horizontal scale in proper distance, shows the lightpear outline. Here is Lineweavers plot of the growth of the scalefactor R(t), which models the growth of all distances between observers at universe-rest (at rest with respect to background.) http://ned.ipac.caltech.edu/level5/M...s/figure14.jpg The dark solid line is according to standard model parameters. Various other cases are shown as well. |
| Aug20-12, 03:48 PM | #412 |
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*Start of the matter era is at radiation/matter equalization around z=3300, with t=0.057 My and Hubble time 0.1 My. This is enough info to bring in the radiation energy effect accurately. |
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| Aug20-12, 05:16 PM | #413 |
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It is intriguing to think of a calculator that you put 3 model parameters into and it then generates a table, going along the t-scale step by step. I would experiment with using the scalefactor as an alternative lefthandcolumn variable, instead of z. The thing is, when someone says we see this galaxy with redshift 4, if you want to look it up you could just think: scalefactor a = 1/(1+z) = 1/5 = 0.2 I am looking at this galaxy as it was when distances were 20% of what they are today. that galaxy I'm looking at is back in the days of scalefactor 0.2 I would want undergrad students to be familiar with converting z that they read into scalefactor, and then putting scalefactor into calculator. So I would put the scalefactor along the lefthand column of the second option table. And next to it the time (derived from that scalefactor). IMO we observe the scalefactor just as directly (from the spectrum of incoming light) as we observe the z. they are just different algebraic versions of the same basic datum. And a is increasing, it is a lot more like t. You've got to follow your own craft-sense. But I think I'll try making a table with increments of scalefactor a and see what it looks like. Maybe it's a bad idea for some reason I don't see yet. Your idea of making something that will accept 3 inputs like (13900, 16300, 0.1 My) and from those 3 inputs crank out a table (even a small table, with specified range and stepsize) is terrific. ================== EDIT: have to go to the trainstation but just want to write down this google code (no time to check it) t+0.001 = (163/1.5)arctanh sqrt(a^3/(a^3 -1 + 1/.7272)) EDIT: google calculator does not have arctanh, or artanh, the inverse of tanh. I will try to implement using the analytical expression for arctanh, which employs the natural logarithm ln(x) t+0.001 = (163/3)ln(1+(1+(1/.7272-1)/a^3)^-.5) - (163/3)ln(1-(1+(1/.7272-1)/a^3)^-.5) When I try this with a = .5 I get that t+.001 = about 59.7. Seems right, so I'll make a table based on the scalefactor. Code:
Scalefactor Age Gy .1 0.56 .2 1.58 .3 2.88 .4 4.37 .5 5.97 .6 7.61 .7 9.24 .8 10.82 .9 12.33 1.0 13.759 1.1 15.11 1.2 16.38 1.3 17.57 1.4 18.70 The table shows that that will happen about 5 billion years from now. |
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| Aug21-12, 12:09 AM | #414 |
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The equation that you use will again be very close for a > .01, but will start to deviate for smaller a, due to the hotter radiation at early times. Our proposed simplified calculator should be fairly accurate for a down to around one millionth or so, provided that we get the radiation component in correctly. |
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| Aug21-12, 12:57 AM | #415 |
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So a is a directly observed (not model dependent) quantity, and it is also a key variable in the calculation. The fact that it's this way gives IMO a solid empirical feel to the situation. I think in my dream calculator you would have a box for 1+z, and a box for a. They are reciprocals of each other, and putting a number in either would work. It wouldn't have a box for z. If a student reads somewhere that a galaxy was observed with redshift 3, then he or she should know to put in 4, or mentally convert that to 1/(3+1) = 0.25 and put 0.25 into the a box. It's getting late here. Maybe some fresh ideas in the morning. I should try to make this more compact: t+0.001 = (163/3)ln(1+(1+(1/.7272-1)/a^3)^-.5) - (163/3)ln(1-(1+(1/.7272-1)/a^3)^-.5) ta = (16.3/3)ln((1+(1+(1/.7272-1)/a^3)^-.5)/(1-(1+(1/.7272-1)/a^3)^-.5)) I've dropped the little time adjustment of .1 million year, and put a decimal point into the 16.3 so it gives answers in billions of years as in that brief table. So now, associated with every directly measurable scalefactor a we have the estimated expansion age TIME when distances were that size, or when the light was emitted. And our handle on how fast the world was expanding at that epoch is the Hubbletime. Basically a sort of linear "doubling time" for distance growth. To every scalefactor a in the past there should be an associated growthrate. Let's add Hubbletime Ya to that brief table: Ya= 16.3(1+(1/.7272-1)/a^3)^-.5 Code:
Scalefactor Age (Gy) Hubbletime (Gy) ? a ta Ya Δa .1 0.56 0.84 5.38 .2 1.58 2.36 3.46 .3 2.88 4.22 2.61 .4 4.37 6.22 2.11 .5 5.97 8.15 1.77 .6 7.61 9.85 1.53 .7 9.24 11.26 1.35 .8 10.82 12.38 1.21 .9 12.33 13.24 1.09 1.0 13.759 13.900 1.00 1.1 15.11 14.40 1.2 16.38 14.77 1.3 17.57 15.06 1.4 18.70 15.29 While I can still edit I'll try adding an interesting incremental distance number that can be calculated at each scalefactor a: Δa = 1/sqrt(.2728*a + .7272*a^2) It may turn out to have no use, but the table has room for another column 
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| Aug21-12, 02:36 PM | #416 |
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The most troublesome one to find from the inputs is the present radiation energy density, but because it is very small (relatively), the following seems to work well: [tex] \Omega_r = \left(\frac{Y_{now}}{Y_{eq}}\right)^2 a^4 - \Omega_m a - \Omega_\Lambda a^4[/tex] where: [itex]a=(1+z_{eq})^{-1}[/itex], [itex] \Omega_\Lambda = (Y_{now}/Y_{inf})^2[/itex], [itex]\Omega_m\approx 1-\Omega_\Lambda[/itex] (provided [itex]\Omega_r \ll 1[/itex]). I have checked this by means of a spreadsheet and it looks promising, with errors far below the input accuracies throughout the redshift range of interest, zero to 3350. Edit: Surprisingly, the rather complex equation may be unnecessary, because a simple [itex] \Omega_r \approx \Omega_m/z_{eq}[/itex] seems to be just as accurate. Marcus mentioned this relationship in a prior reply. |
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| Aug21-12, 02:54 PM | #417 |
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Jorrie, I didn't see your last post (#416) until just now when I posted mine. I'm glad to hear this, it's looking good!
It's intended to be the 13.9/16.3 model we've been concentrating on and I think the first three columns are right, but am not sure about the accuracy of the last two, the distances to a source at the given scalefactor. EDIT: For clarity I will write out the google calculator expression for ta in LaTex: [itex]t_a = \frac{16.3}{3}ln\left((1+(1+(1/.7272-1)/a^3)^{-.5})/(1-(1+(1/.7272-1)/a^3)^{-.5}) \right) [/itex] Here's the expression as used in the calculator: ta = (16.3/3)ln((1+(1+(1/.7272-1)/a^3)^-.5)/(1-(1+(1/.7272-1)/a^3)^-.5)) Here's the expression for the Hubble time: Ya= 16.3(1+(1/.7272-1)/a^3)^-.5 Code:
Scalefactor Age (Gy) Hubbletime (Gy) Proper distance to source (Gly) a 1/a-1 ta Ya Dnow Dthen .1 9.0 0.56 0.84 30.9 3.09 .2 4.0 1.58 2.36 24.0 4.79 .3 2.333 2.88 4.22 18.7 5.62 .4 1.5 4.37 6.22 14.5 5.79 .5 1.0 5.97 8.15 10.9 5.45 .6 0.666 7.61 9.85 7.9 4.74 .7 0.428* 9.24 11.26 5.4 3.78 .8 0.25 10.82 12.38 3.3 2.63 .9 0.111 12.33 13.24 1.5 1.36 1.0 0.0 13.759 13.900 0.00 0.00 1.1 15.11 14.40 1.2 16.38 14.77 1.3 17.57 15.06 1.4 18.70 15.29 (13.9 Gy, 16.3 Gy, flat) → (70.3463, 0.7272, 0.2728) |
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| Aug21-12, 04:06 PM | #418 |
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I also like the scalefactor "input column", because one can go as far into the future as desired ( a > 1). If we use redshift, it would be negative for the future and that's an awkward concept. PS: look at the edit I've made to post #416. |
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| Aug22-12, 03:50 AM | #419 |
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I found out a minor detail about Wright's calculator. when you tell it .7272 and .2728 it actually uses those values, although it REPORTS that it is using ..727 and .273.
IOW it rounds off what it says the model parameters are that it is using, but you can see the difference in the results. It's just a minor thing, but it's convenient. You can actually get that calculator to use (70.3463, .7272, .2728) even though it may look as if you can't (because of this rounding off.) I saw the edit in #416, thanks for the mention :-) it makes sense. That aspect (getting the right radiation component) looks very hopeful. What I'm not sure about is how you will be able to build a different "front end" EDIT: For clarity I will write out the google calculator expression for ta in LaTex: [tex]t_a = \frac{16.3}{3}ln \frac{1+(1+\frac{1/.7272-1}{a^3})^{-.5}}{1-(1+\frac{1/.7272-1}{a^3})^{-.5}} [/tex] [tex]t_a = \frac{16.3}{3}ln \frac{1+(1+(1/.7272-1)/a^3)^{-.5}}{1-(1+(1/.7272-1)/a^3)^{-.5}} [/tex] [itex]t_a = \frac{16.3}{3}ln\left(\frac{(1+(1+(1/.7272-1)/a^3)^{-.5})}{(1-(1+(1/.7272-1)/a^3)^{-.5})} \right) [/itex] [itex]t_a = \frac{16.3}{3}ln\left((1+(1+(1/.7272-1)/a^3)^{-.5})/(1-(1+(1/.7272-1)/a^3)^{-.5}) \right) [/itex] Finally, here's the expression I paste into google calculator for ta: (16.3/3)ln((1+(1+(1/.7272-1)/a^3)^-.5)/(1-(1+(1/.7272-1)/a^3)^-.5)) Here's the corresponding expression for the Hubble time Ya: 16.3(1+(1/.7272-1)/a^3)^-.5 |
| Aug22-12, 09:29 AM | #420 |
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[itex]\Omega_\Lambda = (Y_{now}/Y_{inf})^2[/itex]; [itex]\Omega_m \approx 1-\Omega_\Lambda[/itex]; [itex]\Omega_r\approx \Omega_m /(z_{eq}+1) [/itex] and [itex]H_0 = 1/Y_{now}[/itex] of course. This we send to the full version's numerical integration module. Strictly speaking, we should also input the cosmic time (t) for r-m equality, but provided we start the integration early enough (well before r-m equality), we can set the starting time to zero.
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| Aug22-12, 12:02 PM | #421 |
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My first reaction is (cheers!) we can throw out my google-calculator approximations for anything like z>> 10. They become too inaccurate for z > 100.
But I don't have microsoft Excel on this computer and have never installed a spreadsheet in my life  (our son, who visits now and then, may help with that). So I don't have much of an idea how the new "front-end" will look and work.I have the XLS file on my desktop, waiting, but so far have only opened it as text. I think I need Excel to open it as an actual spreadsheet. This is exciting, I picture that the three inputs to the front end are (ynow, y∞, zeq) and that it outputs perhaps single values of stuff (like a, t, z, ya or yt, Dnow, Dthen...) Or perhaps, if not now then possibly in future, a table. Assuming the user has specified a sequence of values of a, or values of t, to run down the first or lefthand column of the table. That seems pedagogically beautiful, to me. It says to the beginner "all you do is specify two percentage growth growth rates of distance: the present and the eventual future one" and the model does the rest. So attention is focused on percentage growth rate instead of "speed". And the cosmo constant is no mystery, but simply manifest in the eventual percentage growth rate. Looking forward to seeing the actual front-end, this is just how I picture it. |
| Aug22-12, 02:33 PM | #422 |
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In the meantime, I'm getting a routine down for setting up the 13.9/16.3 model in Jorrie's calculator.
First go to google and say Mpc/(km/s)/13.9 billion years that outputs the number 70.3463274so you copy to clipboard and paste into the calculator's Hnow box. Next go to google and say 1 - .7272 - .0000812 that outputs the number 0.2727188so you copy that to clipboard and paste in the Omega Matter Now box. Then you are ready to go! You have almost exactly the right parameters loaded for the 13.9/16.3 model. And if you want to try variations with different ynow and y∞ you can use this same format. The only difference is the second step you say 1 - (ynow/y∞)2 - .0000812 Because all the .7272 is is the square of the ratio of the two Hubbletimes. ====================== Extra, probably unneeded explanation: the second step is to ensure perfect flatness. Premultiplying 1/(13.9 billion years) by the factor Mpc/(km/s) is simply to get rid of the units km/s/Mpc so that what you get out is a pure number 70.346... to paste into the H box. In this approach 1/(13.9 billion years) actually is the Hubble growth rate, expressed in "per time" terms. |
| Aug22-12, 03:38 PM | #423 |
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In the meantime I have attached the spreadsheet data as a .pdf, where you can see the 'front-end', but unfortunately not manipulate it. I could not get my pdf writer to print headers on each page; sorry about that, but it will give you a good idea. The spreadsheet is primitive, but you will see some pointers like "equal", CMB for the green hi-lighted rows... PS: I have also attached the graph for some values on the sheet. |
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| Aug22-12, 04:28 PM | #424 |
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| Aug22-12, 04:56 PM | #425 |
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More to my taste too. So be it.
The graph is handsome. Nice to see the Hubbleradius and the CEH coverging at 16.3 but having a temporary gap. Beautiful curves. Also the red curve, if you flip the graph over, exchanging x and y axes, so time is on the horizontal and scalefactor is up the side, you get this Lineweaver figure #14 http://ned.ipac.caltech.edu/level5/M...s/figure14.jpg The nice thing is you can see the inflection in the curve, where it changes from convex to concave: from decelerating to accelerating. So in your figure you can see the same inflection, a bit below the 10 billion year line, we know it is about 7 billion, but it's nearly linear for a stretch so it's hard to spot the exact point of inflection by eye. |
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