Cosmo calculator-recession speed tutorial

[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 $70 \pm 5$c 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

 

[Jorrie]

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.

What 'discrepancy' are you referring to?
AFAIK, redshifts are measured pretty accurately, but there are uncertainties in the distances to the SN1Ae used for calibrating the 'distance ladder' and hence in the value of H_0. I fail to see how knowing the Doppler shifts will improve that.

 

[JArnold]

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

Marcus, I think I understand the language problem. "Doppler" effects are treated by convention in cosmology as shifts in wavelength produced by local peculiarities in relative motion that can be discounted in measuring cosmological redshift. I should have been clear that I'm referring to doppler redshift in the generic sense: Redshift in wavelength due to the recession velocity of the source, in contrast to redshift due to subsequent cosmological expansion.

Here's the problem as I see it. Relative velocity would, in principle, be a good measure of cosmological distance, but when it's derived from wavelengths, being derivative, it's prone to confusion and miscalculation. 1) When wavelength rather than velocity is used to calculate z, it's evident that recession velocity shouldn't be relativized, because cosmic expansion isn't relativistic. (It's commonly recognized that recession velocities can exceed c, and yet high-z is calculated relativistically.) In the measure of z in terms of the ratio of wavelength-then to wavelength-now it's clear that there's no relativistic limit that would diminish higher ratios, because space and recession velocities can, in principle, expand without limit. 2) Basing z on the ratio between wavelengths brings the problem that I've been struggling with: It doesn't distinguish the redshift due to cosmic expansion from the redshift due to the recession velocity of the source. Consequently, deriving distance from velocity and z as it's constructed only masks that fundamental problem. There must be a unique solution, given the usual parameters (age of universe, Hubble, etc), to discriminate the components of redshift (recession speed and cosmic expansion), but I've been unable to develop it.

 

[JArnold]

What 'discrepancy' are you referring to?
AFAIK, redshifts are measured pretty accurately, but there are uncertainties in the distances to the SN1Ae used for calibrating the 'distance ladder' and hence in the value of H_0. I fail to see how knowing the Doppler shifts will improve that.

As I'm sure you know, the 2011 Nobel was awarded to Riess et al for showing that the universe is accelerating, based on data that shows type 1A supernovae are, for example, about 25% fainter at z=.5 than redshift would indicate.

 

[Jorrie]

As I'm sure you know, the 2011 Nobel was awarded to Riess et al for showing that the universe is accelerating, based on data that shows type 1A supernovae are, for example, about 25% fainter at z=.5 than redshift would indicate.

Yes, but that was not because the redshift was measured incorrectly; it was because the models for converting redshift to distance were based on a non-accelerating cosmos (Lambda=0). By other means, not redshift, they found that those galaxies were farther (dimmer) than previously calculated by the (then) standard model. This essentially increased Ho and required Lambda to be greater than zero, otherwise the models did not fit all observations.

On your problem mentioned to Marcus (which he no doubt will explain in detail): "There must be a unique solution, given the usual parameters (age of universe, Hubble, etc), to discriminate the components of redshift (recession speed and cosmic expansion), but I've been unable to develop it".

You cannot use both expansion factor and recession speed in the calculation for distance, because they are just different views of the same thing (dependent variables). One can say that essentially recession speed is the apparent rate at which the proper distances between us and distant galaxies are increasing, which can exceed c. We should not apply the Doppler shift formula (relativistic or otherwise) to this speed.

 

[JArnold]

Where has it been agreed that Ho should be increased?

Regarding your comment about expansion and recession, I don't know to express my point more clearly than this: At the moment of emission, light can be highly redshifted due to the recession velocity of the emitter. SUBSEQUENTLY, depending on the time it takes for the light to be received, it will be redshifted due to cosmic expansion. The two bases of redshift are separate and independent.

 

[Jorrie]

Where has it been agreed that Ho should be increased?

Ho was still around 50 km/s/Mpc in the mid 1990s; today it is around 70 km/s/Mpc. It was the discovery of accelerating expansion that forced the increase.

Regarding your comment about expansion and recession, I don't know to express my point more clearly than this: At the moment of emission, light can be highly redshifted due to the recession velocity of the emitter.

No, light can only be highly Doppler-shifted if relative to its local area, the source has an extreme peculiar (non-Hubble) recession velocity, causing the Doppler shift. Galaxies typically do not have that; even adjacent clusters move relative to each other at no more than around c/1000, as Marcus also stated. In a z=0.5 galaxy, that accounts for a negligible amount. What we observe is all due to expansion, unless that galaxy is very nearby - like in the Virgo cluster, where we do not apply Hubble's law.

It is possible to approximate the cosmic redshift by a series of infinitesimal Doppler shifts between adjacent points in the line of sight, but AFAIK, that's not a common method any more. Take note that in such a case, the expansion factor is not used, so that we do not double-count.

 

[JArnold]

Jorrie, thank you for your patience. I was wrong, you were right. What finally made sense to me was a thought experiment (in the shower!): If the universe were to abruptly stop expanding, would a distant galaxy still recede, and have a redshift? The answer of course is no.

 

[Jorrie]

If the universe were to abruptly stop expanding, would a distant galaxy still recede, and have a redshift? The answer of course is no.

Yes, showers do some magic, sometimes...

But, note that such a case would have made no difference to our present redshift observation of distant galaxies; we would have to 'wait for billions of years' to notice the change. What we currently observe in terms of redshift is determined only by how much the universe has expanded since the time of the emission of those photons.

In any case, this is another (good) argument against Doppler shift due to recession velocity, at least as per the cosmological understanding of the terms. One can obviously also just look at the good old balloon analogy and "see the light"...