Cosmo calculator-recession speed tutorial

[stevebd1]

Thanks for the reply, Jorrie. I've plugged the revised quantities of omega and left them as unitless fractions of the critical density into the equation but the answer still seems incorrect.


\Delta t = \frac{\Delta a}{H_0\sqrt{\Omega_m/a^3+\Omega_r/a^4 + \Omega_\Lambda}}


\Delta t = \frac{-7.8\text x10^{-5}}{0.07256\sqrt{\frac{0.27}{0.000847^3}+\frac{0.000084}{0.000847^4} + 0.73}}


\Delta t = \frac{-7.8\text x10^{-5}}{0.07256\sqrt{\frac{0.27}{6.0765\text x10^{-10}}+\frac{0.000084}{5.1468\text x10^{-13}} + 0.73}}


\Delta t = \frac{-7.8\text x10^{-5}}{0.07256\sqrt{0.0444\text x10^{10} + 0.1632\text x10^{9} + 0.73}}


\Delta t = \frac{-7.8\text x10^{-5}}{0.07256\sqrt{6.072\text x10^{8}}}


\Delta t = \frac{-7.8\text x10^{-5}}{0.07256\,\text x\,2.464\text x10^{4}}


\Delta t = \frac{-7.8\text x10^{-5}}{0.1788\text x10^{4}}


\Delta t = -4.362\text x10^{-8}


It would be interesting to know exactly what units the answer is supposed to be expressed in. I'm beginning to think I might have something amiss with the quantities for a and \Delta a. Also, having tried other z-parameters, is it the norm that \Omega_\Lambda contributes so little to the calculations?

regards
Steve

 

[marcus]

Here's a practical question I'd appreciate some help with if not too much trouble.
Jorrie, Hellfire, Steve, you all have calculators or programs for calculating this kind of thing and I'm concerned with the range of variation

I want to know the recession speed of z = 1100 matter at the time the light is emitted.

I want to use fairly standard parameters like what is default in Wright's calculator:
Hubble 71
Omega_matter 0.27
Omega_lambda 0.73

If I use those parameters and put z = 1100 into Morgan's calculator, it says that the recession speed (back then when the light was emitted) was 57c.

what I'd like to know is, what do you all with your different calculators get instead of 57c. Or do you get that same answer more or less exactly.

What we are talking about (in case someone reading the thread wonders) is the recession speed of the last scattering surface at the time when the CMB light that we are now receiving was emitted.

 

[Jorrie]

\Delta t = -4.362\text x10^{-8}

It would be interesting to know exactly what units the answer is supposed to be expressed in. I'm beginning to think I might have something amiss with the quantities for a and \Delta a. Also, having tried other z-parameters, is it the norm that \Omega_\Lambda contributes so little to the calculations?

Hi Steve. Working through your calcs, I spotted an error in the equation that I gave! It should be:

\Delta t = \frac{\Delta a}{aH_0\sqrt{\Omega_m/a^3+\Omega_r/a^4 + \Omega_\Lambda}}

Note the subtle aH_0 below the line. I accidentally omitted the a in that formula. Sorry about that! This would place your result in the right ballpark.

BTW, if you express H_0 in Gy^{-1}, then \Delta t must be in Gy.

Regards, Jorrie

 

[Jorrie]

If I use those parameters and put z = 1100 into Morgan's calculator, it says that the recession speed (back then when the light was emitted) was 57c.

what I'd like to know is, what do you all with your different calculators get instead of 57c. Or do you get that same answer more or less exactly.

Hi Marcus. I find the differences to originate with the assumptions for the radiation energy density parameter today. Morgan's calculator ignores it, I use 8.35\times 10^{-5} and Hellfire's calculator uses 0.0005\Omega_m.

As a test, I plugged \Omega_r=0, \Omega_r=8.35\times 10^{-5} and \Omega_r=1.35\times 10^{-4} into my spreadsheet and got the following results for the recession speeds at z ~ 1100 respectively: d(aR)/dt= 57c, 66c and 71c, which is what the respective calculators yield.

Different radiation densities, different results in the very early epochs...

Regards, Jorrie

 

[marcus]

...Morgan's calculator ignores it, I use 8.35\times 10^{-5} and Hellfire's calculator uses 0.0005\Omega_m.

Thanks Jorrie! I had in mind a rough magnitude estimate that turns out similar to what Hellfire has----I was thinking that today's radiation density is roughly 1/2000 of today's matter density. the radiation is almost all CMB, I guess. That is the same as the 0.0005 Hellfire is using. I can't tell you the source though.

As a test, I plugged \Omega_r=0, \Omega_r=8.35\times 10^{-5} and \Omega_r=1.35\times 10^{-4} into my spreadsheet and got the following results for the recession speeds at z ~ 1100 respectively: d(aR)/dt= 57c, 66c and 71c, which is what the respective calculators yield.

Do you have a link to anybody's energy density inventory? I saw one on arxiv several years back but it would be bother to hunt down again. It would be nice to know the CMB energy density. I guess one could get it from the temperature by a version of Stefan-Boltzmann. I'n lazy as usual.

I want to give someone an example. Maybe I will just say "about 70c"
that would agree roughly with both your figure and Hellfire's, would it not?

============
Argh. I bit the bullet and went and hunted up the inventory I saw earlier and it is 2004 by Peebles
http://arxiv.org/abs/astro-ph/0406095
and it says the Omega_radiation is about 10^-4.3
that is 5E-5, or 1/20,000

Hey Jorrie! that 5E-5 is very close to your 8.35E-5
I consider it very likely that you have a better source than I do----better than the Peebles article.

 

[stevebd1]

\Delta t = \frac{\Delta a}{aH_0\sqrt{\Omega_m/a^3+\Omega_r/a^4 + \Omega_\Lambda}}

Hi Jorrie, as you probably already know, that works out. The answer I get is -0.000051504 Gyrs which is -51,504 years, close to what it was estimated to be (-53,000 yrs). As you previously stated, \Delta z = 10 would probably give more accuracy.

Steve

 

[hellfire]

It is already a long time ago since I implemented my calculator and the Omega radiation seems to be obsolete. Jorrie's value seems to be according to the WMAP estimations and I think it should be taken as the correct one. By the way, the accuracy of those calculators for high redshifts should not be overestimated. Even for some cosmological models the deviations are too big due to the finite integration steps. For example, for an eternal de-Sitter model (no matter density and cosmological constant = 1) all calculators predict a finite age of the universe. May be it is also worth to mention that Ned Wright's and Siobahn Morgan's are implemented according to the same source code that requires at least two integration loops. Mine, however, is based on a different more simple approach with one single loop integrating from a = 0 to a = 1 and getting the required values at a = a(z). The code can be found at:
http://www.geocities.com/alschairn/cc_e.js

 

[Jorrie]

Do you have a link to anybody's energy density inventory?

Hi Marcus, I could not quickly find a link, but I think it is the 3-year WMAP data analysis that supported that figure of mine, as Hellfire mentioned. The difference is just about 50% and it makes some difference in the very high redshift epochs. However, other uncertainties also exist, like Ho which is still uncertain with +-3 Km/s/Mpc, I think!

I want to give someone an example. Maybe I will just say "about 70c" that would agree roughly with both your figure and Hellfire's, would it not?

Yep, I guess 70 \pm 5c is probably as good as we can state today.

Jorrie

 

[Jorrie]

For example, for an eternal de-Sitter model (no matter density and cosmological constant = 1) all calculators predict a finite age of the universe.

Do you mean the 'eternal inflation' model? If so, isn't this to be expected for a non-zero Ho?

For zero Ho, most calculators will 'blow up', correctly suggesting a tendency to infinite age.

Thanks for the link to your code, Hellfire!

Jorrie

 

[stevebd1]

Jorrie, thanks for answering all my queries. I'm under the impression there are a whole set of equations for working out the various quantities indicated in the calculators such as 'distance then', 'distance now', 'Hubble parameter then' etc. Is there somewhere online where these equations are shown (particularly regarding how the Hubble parameter then and proper distances now & then are calculated).

I'd also be interested to know what R represents in the equation below and how it contributes to the calculation of the recession speed-

As a test, I plugged \Omega_r=0, \Omega_r=8.35\times 10^{-5} and \Omega_r=1.35\times 10^{-4} into my spreadsheet and got the following results for the recession speeds at z ~ 1100 respectively: d(aR)/dt= 57c, 66c and 71c, which is what the respective calculators yield.

regards
Steve

 

[hellfire]

Do you mean the 'eternal inflation' model? If so, isn't this to be expected for a non-zero Ho?

I mean a de-Sitter cosmological model with a ~ exp(Ht), like for example the steady-state cosmology:
http://www.daviddarling.info/encyclopedia/D/de_Sitter_model.html

 

[Jorrie]

Is there somewhere online where these equations are shown (particularly regarding how the Hubble parameter then and proper distances now & then are calculated).

Hellfire's code (link in https://www.physicsforums.com/showpost.php?p=1575704&postcount=37") gives a good idea of how those values are calculated.

I'd also be interested to know what R represents in the equation below and how it contributes to the calculation of the recession speed.

My R is the present proper radius of the observable universe (~46 Gly), which is also approximately the proper distance to the region that emitted the CMB that we observe today. When multiplied by the expansion factor a, it gives the proper radius of that region at any past epoch. The rate of change of aR, i.e. d(aR)/dt, is the recession speed that Marcus was talking about.

Jorrie

 

[Jorrie]

I mean a de-Sitter cosmological model with a ~ exp(Ht), like for example the steady-state cosmology...

Does it mean that the de-Sitter expansion (a ~ exp(Ht)) started effectively an infinitely long time ago and hence the age at any time must be infinite?

 

[hellfire]

Does it mean that the de-Sitter expansion (a ~ exp(Ht)) started effectively an infinitely long time ago and hence the age at any time must be infinite?

Correct. Such a model has no initial singularity and the perfect cosmological principle applies.

 

[Jorrie]

Hi Hellfire, thanks.

I've looked at your http://www.geocities.com/alschairn/cc_e.js"!

 

[jonmtkisco]

By the way, the accuracy of those calculators for high redshifts should not be overestimated. Even for some cosmological models the deviations are too big due to the finite integration steps.

Hi Hellfire and Jorrie,

I think that the problem with the finite integration steps is exactly why I got large deviations when I tried to start from close to t=0 and integrate forward to the present. I got discouraged at the prospect for reconciling the differences, and if they can't be reconciled, it makes the forward calculation pretty useless. Even though in theory it ought to be possible to calculate it correctly -- at least from after the first 2 seconds or so.

Jon

 

[Jorrie]

Hi Hellfire and Jorrie,

I think that the problem with the finite integration steps is exactly why I got large deviations when I tried to start from close to t=0 and integrate forward to the present.

Yea, when you are interested in the first few seconds/minutes after the end of inflation, you may have problems, but I think our simple calculators can be fairly accurate from about z = 1,000,000 onwards, corresponding to an age of around 1 year max, which you can just as well make zero on cosmological scales!

Hellfire's calculator and my spreadsheet both give a recession speed for an object at a hypothetical z=1,000,000 as ~30,000c, at a distance "then" of ~50,000 ly. A constant recession rate will give the expansion time as ~1.6 years and the actual time must be much less.

Jorrie

 

[marcus]

This is a thread from earlier this year with some links to cosmology resources----calculators in particular. Came across it by accident today. Maybe we need some stuff about cosmology calculators. They embody the standard model in a hands-on way.
Some PF people have developed their own, based on spreadsheets I think.
Are there any new ones anybody knows of? Recent experience to share?
Improved versions?

 

[simple1212]

wonderful post.. great going..

http://simpleinterestcalculator.org [/color]​

 

[marcus]

Simple1212, thanks for the encouragement! and welcome to Physicsforums. I'm glad to know this thread has been useful to you.

The link in the very first post is old and draws a blank. Later that year, Morgan changed the link to the calculator to:
http://www.uni.edu/morgans/ajjar/Cosmology/cosmos.html

That link has worked for over two years now. The calculator is basically real easy to use, but everyone feel free to ask if you find it acting cranky or giving weird results.

 

[marcus]

If anyone wants to work out a couple of recession speeds just to keep in practice.
The most distant OBJECT observed so far is the giant star at z = 8.3 that produced GRB 090423.

That was the big gammaray flash seen in April 2009.
Here is technical detail if anyone wants:
http://arxiv.org/abs/0902.2419

There is a great Perimeter video seminar talk just recently posted that discusses how they measure the redshift of these things and how they think the flashes are produced---by an unusual type of supernova of a rapidly spinning giant star where more happens than just the usual supernova mechanisms.

Or else by the abrupt merger of two compact objects like two neutron stars (abbreviated NS-NS)
or NS-BH. Eliot Quataert gives the talk.http://pirsa.org/09090028/

I'd say forget the technical paper, the Perimeter video is so good. A lot of the images are animated. The presenter, Eliot, is excellent. The latest understanding on how GRBs work.

Anyway try calculating the recession rate at the time the flash occurred, when the gamma started on its way to us. And also calculate the recession rate of the dead star remmant now today as the gamma arrives here. And the present distance.

What I get is that the expansion was 620 million years old when the flash occurred. And its age is about 13.7 billion years now. So the light has been traveling about 13 billion years.
See what you get. I used the old numbers: 0.27, 0.73, and 71 for matter fraction, cosmo constant, and present Hubble rate.
Some people might prefer the newer 0.25, 0.75, and 74. But it won't make a lot of difference, anything roughly around those values works OK.

What do you get for the two recession rates? the "now" and the "then" rate that the distance to the star was increasing.To repeat, this is the most distant object yet observed. The CMB is glow from hot gas, so not really an object. That hot gas is the most distant material observed--- redshift z = 1090---but the star that produced GRB 0904023 at z = 8.3 is the most distant condensed object.

 

[stevebd1]

I'm aware that this is an old thread but due to the fact that Hellfire's cosmic calculator, which included for Omega radiation, is no longer available, I thought I'd add a link to the calculator that Jorrie put together based on hellfire's calculator-

http://www.einsteins-theory-of-relativity-4engineers.com/cosmocalc.htm

 

[JArnold]

Seems to me an accurate value for z would have to include both the cosmic expansion during the light travel time AND the doppler of the source at the moment of emission.

 

[marcus]

Seems to me an accurate value for z would have to include both the cosmic expansion during the light travel time AND the doppler of the source at the moment of emission.

Definitely! In many cases (essentially nearby stuff) the local motion doppler can be figured out!

Individual motion of galaxies tends to be on the order of 300 km/s (very roughly) or about 1/1000 of speed of light.

So the doppler effect on z would be something on the order of 0.001.

You can see that for the vast majority of cases (say z > 0.1) the doppler caused by the individual motion of the source is not going to matter.

But on the other hand for very nearby galaxies, their cosmological z is essentially nil. Their individual motions relative to us are the only thing that counts.

 

[JArnold]

Is there a calculator for high-z that incorporates both?

 

[marcus]

No, I wouldn't think so. Local motions of galaxies are in pretty much random directions. For high z you can neglect them, the doppler effect would be less than the margin of error. I don't see how one could construct such a calculator.

 

[Jorrie]

Is there a calculator for high-z that incorporates both?

I don't think so, because the Doppler shifts are swamped by the high-z's. Another problem is that we do not even know if the peculiar motions of distant galaxies are positive or negative relative to us, so how would one include them in a calculator?

 

[marcus]

Good point! We posted simultaneously it looks like. I agree with Jorrie.
(I keep the link to your calculator in my signature now. It's a good one.)

 

[JArnold]

But a galaxy with a cosmic z of, say, 1 would have had a significant doppler at the time of emission. One of the points of a paper I'm preparing is that it might account for the discrepancy between type 1A supernovae distances and redshifts.

 

[marcus]

But a galaxy with a cosmic z of, say, 1 would have had a significant doppler at the time of emission. One of the points of a paper I'm preparing is that it might account for the discrepancy between type 1A supernovae distances and redshifts.

I see now. There is a serious misunderstanding of terms. You and I are speaking a different language, in effect.

Here's how I talk:
The doppler associated with random local motion would typically be on the order of 0.001 or less and equally likely to be a blueshift as a redshift. One cannot predict for a general galaxy at z=1 what that individual motion, and that doppler, would be. But it is negligible compared with the cosmic redshift z = 1.

Here's how you talk:
A galaxy with cosmic redshift z=1 would have [in addition to that?] a significant [?] doppler [resulting from what?] at the time of emission.

I think one could say that you are double counting. The cumulative effect of all the cosmological expansion the light experiences at the time of emission and at all the other times the light is traveling is already taken account of in the redshift z=1