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Relativity and Cosmology

This is a Blog on relativity and cosmology for engineers and the like. My website "Relativity-4-Engineers" has more in-depth stuff.

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Cosmology Equations Part 5

Posted November 12, 2006 11:00 AM by Jorrie

In Cosmology Equations Part 4 we looked at the famous Friedmann equation, for which some applications will be discussed in this installment. First the equation again:

where a is universal expansion factor, a-dot = da/dt the expansion rate, H0-bar the normalized Hubble constant, Ω and its Ωi components are the universal energy density parameters, referring to: total energy density, mass density, radiation density and vacuum energy density.

The first exercise is to plot the expansion factor against time. For this we must integrate the expansion rate a-dot against time. Sadly, this is not possible analytically. Happily, we can do it numerically! Engineers do this a lot when they simulate complex dynamic systems. So, without wasting more time, let's plot the expansion curve (a link to the spreadsheet responsible for this will be given later).

The values used are: Ωm = 0.27, Ωv = 0.73, Ωr = 0, H0 = 70 Km/s/Mpc. The vertical blue line at 13.7 Gy represents "now", with values to the left of it representing the past and values to the right of it the future (which is only shown so that the increasing expansion rate becomes more visible).

The horizontal red line at redshift 1.6 illustrates one of the useful applications of this graph: the length of the line, between "now" and the redshift curve, represents the look-back time to objects observed at redshift 1.6, which in this case is 9.7 Gy. It means that the light from a galaxy that we today observed at redshift 1.6, took 9.7 billion years to reach us.

Looking more closely at the blue expansion curve, note that during the first 5 Gy or so, the expansion rate slowed down under the dominance of matter density (Ωm). Radiation density is ignored, because it was only the first thousand years or so that it played a role and that's insignificant on this giga-year scale.

Then there was a period where matter density and vacuum energy density (Ωv) more or less balanced each other - the cosmic 'middle ages' - where the curve is almost straight. From about 10 Gy onwards, it is clear that the expansion rate is increasing under the dominant vacuum energy. This is what we observe today.

The look-back redshift (z) curve simply represents the inverse of the expansion factor (actually z+1 = 1/a). The curve cuts the time axis at t = 13.7 Gy (now) and stops there, because there is no point in considering redshift into our future! To the left, the curve rises to approach infinity at the time of the Big Bang, but is cut off at z = 2 for no other reason than to get good resolution on the graph at the lower values of a and z.

If you want to, you can download the EXCEL spreadsheet used to plot the curve and play around with the values (best is to right-click the link and select "save target as"). The numeric integration shown is of t against a, i.e., t = [integral f(a) da] for a from the 10-49 to 2, because it works better for the near-infinite slope of the expansion curve when t -› 0 (the time after inflation is about 10-49 Gy).

Note that this graph ignores the inflationary epoch, because it's utterly negligible and invisible on a linear scale graph over the times scales considered. According to our best theory, it lasted from about 10-34 to 10-32 seconds after the Big Bang. The next part of this mini-series will deal with the cosmic inflation epoch - and even earlier!

More about the Friedmann equation and its applications are available on the website Relativity 4 Engineers.

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#1

Re: Cosmology Equations Part 5

11/12/2006 12:05 PM

Hi Jorrie, good article(s), but my head hurts trying to get it around your parameter "a", the "expansion factor". Reading your parts 4 and 5, I still don't get it! Maybe if you can give us a relationship between "a" and the size of the universe?

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#2
In reply to #1

Re: Cosmology Equations Part 5

11/12/2006 12:34 PM

Guest asked: "Maybe if you can give us a relationship between "a" and the size of the universe?"

The size of the universe is not known to us and may never be. What we have a reasonably good value on is the size of the observable universe at the present time. This is usually given in 'light-travel-time' distance, which has the same value as the age of the universe, presently thought to be 13.7 billion years.

While a is just a ratio between the amount of expansion at any time and the amount of expansion today, we can use it to express the size of the observable universe at any time. How? From the expansion curve, get the value of a for any time and multiply it by 13.7. This gives the radius of the observable universe in Gly (Giga, or billion light years) for that time. So...

R(t) = a(t) Ro

where R(t) is the time variable radius, a(t) the time variable expansion factor and Ro the present radius of the observable universe. Straight and simple. Hope it helps with the headache!

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#3
In reply to #2

Re: Cosmology Equations Part 5

11/13/2006 2:47 AM

Tx Jorrie, makes more sense to me now. Will probably be back with more questions later.

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#4

Re: Cosmology Equations Part 5

11/13/2006 11:37 AM

Hi Jorrie, with my 'tongue slightly in cheek' may I postulate the 'Tired Light' hypothesis to explain 'Red Shit' now that new data suggests a way round the 'scattering problem' we could be in for a 'Devil's Advocate' conference?

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#5
In reply to #4

Re: Cosmology Equations Part 5

11/13/2006 1:28 PM

Hi Alastair, if we 'rejuvenate the light' and the 'red shit' becomes 'blue shit', it may become a 'convergence problem' and we may be in for one 'hell of a conference'!

BTW, the Andromeda Galaxy threatens the Milky Way with a blue shift and a possible future collision! I suppose, 'shit does happen!'

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#6
In reply to #5

Re: Cosmology Equations Part 5

11/13/2006 3:00 PM

Jorrie, I think myself that 'Tired Light' is probably dead and buried, but these ideas are a bit like the 'Dracula' movie, He gets the point at the end of one film, only to return in a later movie. I rather like the 'Saddle' postulate, as I was brought up with The Lone Ranger and Tonto. They were like G-d to me. I am sure if you looked closely at the saddle it would look pitted just like leather, except there would be black holes in the wells of those pits. 'Devadata' is the name of Kalki's horse. another name for Lord Shiva, in the Hindu Pantheon. Eastern concepts of the cosmos are quite compatible with modern science, i.e. 4.32 Billion Years ago, is given as the start of this Day of Brahma, close to the accepted age of the planet. The good news is that we exhaust all available material in the universe, constructing a huge pallace around Brahmaloka, to enjoy a big party for 4.32 billion years of 'night' then start all over. for 101 years and a day (360 day years) It all comes to Pi. Mythology perhaps, but with mathematics thrown in, I suspect ancient sages were worried books might get destroyed, but tales round the camp fire survive.

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#7
In reply to #6

Re: Cosmology Equations Part 5

11/19/2006 12:10 PM

Fascinating myth, Alastair, thanks for sharing it!

Also fascinating is that Earth formed shortly after dark energy took over the cosmic expansion dynamics - or is that rather scary?

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#8
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Re: Cosmology Equations Part 5

11/21/2006 6:48 PM

Jorrie, 'Dark Energy', Is that the realm of 'Darth Vader'? I shall keep my 'Light Sabre' charged up. The Hindu pantheon of gods is much like a Russian Doll, layers and layers of meaning within meaning. They are intentionally like Fairey Tales, as otherwise it was claimed the essence would be lost, bedtime storeys tend to pass down from generation to generation largely intact.

The Vedic 'Myths' are quite fascinating, it is repoted that a very highly advanced civilization existed. One claim was that plants were bio-engineered to act as surrogate mothers. In 1967 I painted a mural of this, and won the College Art Prize with it. Both grandfathers served in India. Pater's as ADC to the Viceroy, and Mater's as CEO of the Co-Op and Tea Planters association. Fermi borrowed Grandpa's name. Francis told me the Manhatten Project (at that Time Grandad was on the Board of British Oxygen) were concerned that a test explosion might ignite the atmosphere. Geological data from India suggested otherwise. Pravda wrote a fine artical on the claimed nuclear detonation over Mohenjo-Daro, Trinity Point was detonated on G-Pa's Birthday.

Check out:- http://english.pravda.ru/science/19/94/377/13920_stones.html

Myths anf Fables, Tales from the camp fire under a canopy of stars. Magical stuff in my view.

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#9
In reply to #7

Re: Cosmology Equations Part 5

12/06/2006 11:15 AM

Hi Jorrie, Seems as if 'Dark Energy' is a hot topic of debate right now, Ten to the power of 34 Joule Seconds for the Dark Energy in the black dregs of my coffee cup is about enough to boil off the entire oceans with plenty to spare. Hope nobody tells Darth Vader how to make a 'Destructor Beam' from it

I liked the mathematics, I love it when it's just basic algebra. Stochastic analysis is starting to worry me a bit.....'what is chaos?' some of the time it's very un-chaotic.

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#10
In reply to #9

Re: Cosmology Equations Part 5

12/06/2006 12:32 PM

Hi Alastair, you remarked on dark energy: "Hope nobody tells Darth Vader how to make a 'Destructor Beam' from it"

Even Darth Vader should be sacred, very scared - dark energy does not have a split personality, it has a triple-personality! It takes on personalities like "phantom energy", "vacuum energy" or "quintessence" just by the 'tweak' of a parameter like w; and that for as little a change of w as from -0.99 to -1.01!

If you are not scared to death yet, read this article!

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#11
In reply to #10

Re: Cosmology Equations Part 5

12/06/2006 7:54 PM

Thanks Jorrie, Just had a wonderful surf, the GP-B Gyros by chance two spinning at near minimum inertial axis and two at max inertial axis may be a happy accident. Inertia and Zero Point 'Dark Energy' are believed to be linked. We may get some unexpected data, with further analysis. (Dr. John P.Boyde...we need your monotonous algorithms)

Bottom Line:-

If you are interested in learning more about polhode behavior, the following references may be helpful:

  1. Donald T. Greenwood. Principles of Dynamics, 2nd Ed. (Prentice-Hall, 1988)
  2. Herbert Goldstein. Classical Mechanics, 3rd Ed. (Addison Wesley, 2002)
  3. Vader D.Quintessential Gyroscopic Balls 5th Ed. (Sith-Lord Coruscant Publications)
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#12
In reply to #11

Re: Cosmology Equations Part 5

12/06/2006 8:14 PM

Postscript:- My Colleague Dr.D.V. wants to know what the 3400th TTW 3450th TTG Precision Measuring Equipment Specialists at L.A.F.B. Colorado make of the viscosity of those 'balls' ? Mmmm. we need it to figure out a nice monotonous algorithm. The Air Force Lads I read are helping with the mission. Cool.

Quote:-"Monotonicity, which has been the holy grail of so much recent algorithm development, is a reasonable goal only for ordinary viscosity. Hyperviscous fronts and shock zones in flows with spectral viscosity are supposed to oscillate." ... I think we can discount Hyperviscous fronts etc.

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#13
In reply to #10

Re: Cosmology Equations Part 5

04/17/2007 8:36 PM

Hi Jorrie,

In your link above it states: "This alternative to the accelerating Universe is still a very good fit to all the supernova data." Now it seems there is doubt about the acceleration. I found this link a bit hard to read. In the graph on the right side with Ωm on the horizontal axis and λ on the vertical, am I to understand that we could have a universe that is closed, but can expand forever?

Regards,

S

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#14
In reply to #13

Re: Cosmology Equations Part 5

04/17/2007 9:37 PM

Thanks, StandardsGuy,

for bringing this thread back to life. I have missed our dear friend Jorrie. In private correspondence, another CR4 Member, revealed that he knows of a fine engineer here who is perfectly happy with the concept of our Universe being contained within a gigantic Iron Sphere.....Terry Pratchett, eat your heart out!

Before we universally adopted the 'Helli-centric' idea of our planetary system, Astronomers made a good living calculating ever more complicated 'epicycles' to account for the observations of planetary regression and the like. Clock-smiths got in on the act as well, also making a tidy income from the nonsense. Nonsense that could lead to imprisonment or burning at the stake, if disputed.

I will look for the link to a rather cunning experiment currently being conducted, The results will not be available for several years. Two large birthday-cake size cylinders of brass are placed in very close proximity. Each cylinder has six large holes, the top cylinder is suspended just above the bottom, by a taught wire. The top cylinder oscillates back and forth, much like those perpetual clocks many of us are familiar with in shop displays. Naturally as the holes coincide every so often, a gravitational 'pico?/femto?/atto? - "jolt"... so to speak... is imparted to the angular momentum. The objective of the experiment, is to slowly lower the top cylinder, and thereby test the ubiquitous Gravitational "Inverse Square Law". There is considerable speculation, supported by observation that this may not be the case. One such anomaly is the position of that satellite we slung out of our solar system, with a gold gramophone record of earth sounds etc. Nobody expected the power supply to last that long, but it has, and what is more, she ain't where she is supposed to be!

Jorry will put us on the right track for sure..

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#15
In reply to #14

Re: Cosmology Equations Part 5

04/17/2007 10:05 PM

Hi Alastair,

I am very interested in that gravitational experiment. Is there a link? Are you saying that there is observational evidence that says the gravitational Inverse Square Law is wrong (other than the satellite)? If so, I would like to know about it. I haven't heard about the satellite being in the wrong place, but there could be another explanation (such as a black hole!). Now there is bound to be more activity here, since I said that.

Regards,

S

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#16
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Re: Cosmology Equations Part 5

04/17/2007 10:38 PM

What a great tag "StandardsGuy",

I will try to search out for it. I first read about it in New Scientist a while back. but those magazines pile up fast, then go to the recycle bin. One dispute involves the speed at which Galaxies have been measured to rotate. As time has passed, and better telescopes have fixed positions more accurately etc. there appears to be an anomaly, they appear to be spinning at the wrong speed for their mass? I suppose that if each Galaxy contained a much bigger 'black hole' in the middle/whatever? than so far estimated, that might? conceivably account for it? but again might pose other objections. I am out of my depth here. Measurement is absolutely fundamental to scientific understanding. New data is always very exciting, if corroborated. A new piece to fit in the puzzle, may require the removal of a wrongly positioned piece? on occasions.

Jorry is more qualified than I am to comment.

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#17
In reply to #16

Re: Cosmology Equations Part 5

04/18/2007 8:26 AM

Hi Alastair and SG,

The issue of gravity working by an inverse square law can been proven to arbitrary accuracy, but AFAIK, only on "ordinary" scales, meaning not on the extremely tiny (microscopic) and the extremely large (cosmological) scales. One thing that scientists agree on is that it either does not work on sub-atomic scales, or the other forces overwhelm it completely.

There are two astronomical observations that might perhaps indicate that it also does not work on the extremely large scales, but the evidence is very, very shaky. The galactic rotation curves that Alastair mentioned might be dark matter or it might be a deviation from the inverse square law. However, other observations tend to point towards dark matter and towards ruling out the MOND (modified Newton dynamics) theory.

The spaceship problem that you mentioned is known as the Pioneer Anomaly. Both Pioneer spacecraft suffered a mysterious deceleration, over and above Newton/Einstein's, after they were past Pluto. However, since it has not been observed on other spacecraft, there is suspicion that it may be a systematic for those two identical craft, e.g., the way their nuclear power plants radiate heat.

So, the jury is still out on some of these issues - I'll be very surprised if Einstein's theory of gravitation is found to be wanting on the very large scales!

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#18
In reply to #17

Re: Cosmology Equations Part 5

04/18/2007 8:48 AM

Hi Jorrie,

It's great to read your words of profound wisdom again, Thanks for clearing that issue up as best it can for the moment. I liked the allusion to the 'extremely tiny'. One thing that puzzles me, is that liquid Helium is a 'super-fluid'....it creeps up the side of the flask, and empties itself on your laboratory floor if you are not careful.

Now here is the puzzle; Helium remains a liquid as far as the current art has allowed it to cool, somewhat less than one millionth of a degree Kelvin above absolute. There the Helium molecules are whizzing around like crazy, if we could watch them at that microscopic scale and low temperature. with no energy to speak of, yet never mind gravity, that superfluid liquid is just itching to mess up the floor by creeping stealthily up the sides of the container. Where is the 'energy' coming from?

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#19
In reply to #13

Re: Cosmology Equations Part 5

04/18/2007 9:07 AM

Hi S, you asked: "In the graph on the right side with Ωm on the horizontal axis and λ on the vertical, am I to understand that we could have a universe that is closed, but can expand forever?"

The way I read that figure with Ωm vs. λ graph is that the best fit for all the data is the purple area, roughly where the yellow and blue ares overlap. This gives Ωm ~ 0.3 and λ ~ 0.7, meaning a flat universe with ~70% dark energy (Ωm includes both dark and ordinary visible matter).

The issue about "This alternative to the accelerating Universe is still a very good fit to all the supernova data" is what it says - the Ω=1 model with evolving supernovae it is only a good fit to the supernovae data, not the WMAP data.

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