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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.

Comments/questions of a general nature should preferably be posted to the FAQ section of this Blog (http://cr4.globalspec.com/blogentry/316/Relativity-Cosmology-FAQ).

A complete index to the Relativity and Cosmology Blog can be viewed here: http://cr4.globalspec.com/blog/browse/22/Relativity-and-Cosmology"

Regards, Jorrie

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Cool Cosmo Calculator

Posted September 18, 2012 11:00 AM by Jorrie

The thinking around cosmological models has undergone a shift during the last two decades. My prior blog post hinted at the importance of the Hubble radius and the 'emergent spacetime' view that is implied.

Models seem to have moved from the Einstein-de Sitter (matter dominated) view, through the ΛCDM (dark energy dominated) decade to the present 'emergent spacetime' views. The ΛCDM model can actually handle all the views, simply by setting its parameters appropriately. My old Cosmo Calculator[1] does exactly that and spews out numerous calculated results for a given set of inputs - actually enough information to confuse everyone, except perhaps the experts.

In line with the new thinking, I have teamed up with Marcus from PhysicsForums[2] to develop a more intuitively simple calculator that may be more appealing at beginners level, while still conforming to the concordant cosmic model of today. It is called CosmoLean and is presently at version A25. 'Lean', because it does not flood the user with results. The main output is a table giving the evolution of some useful parameters over time. Before you dig into it, please read on to get some perspective on it first.

CosmoLean starts with a premise that we can forget about dark energy if we accept that there exists an underlying constant spacetime curvature, but that space is perfectly flat.[3] Without going into the "why is this so?" type questions, let me just say that it is perfectly in line with all present observations and paints a picture that is compatible with, but conceptually simpler than the dark energy paradigm. In effect it says that from Einstein's 1915 papers[4] onwards, the laws of gravity had two constants: the local scale gravitational constant (G) and the large scale cosmological constant (Λ), both representing a curvature of spacetime, as applicable to their respective regimes.

Without going into this formidable equation, let's move on to what we need for a new calculator. As in my prior Blog, what we observe today is a (sort of) transient to a future constant Hubble radius (RH), which only depends on the cosmological constant Λ. How close we are to this future constant (say RH_inf) depends on how much the radiation and matter energies have been diluted by the cosmic expansion. We can completely specify this by three parameters, RH_now, RH_inf and the ratio between matter and radiation energy density. For the latter we can choose the redshift at which they were equal (zeq) in the past.

RH_now = cTH_now , the speed of light multiplied by the Hubble time. This again is inversely proportional to the Hubble constant (H0). If we choose units appropriately,[5] then c=1 and TH = 1/H0, which has the value 13.9 Gy, according to today's best observations. The future value RH_inf =16.3 Gly and the past value zeq ~ 3500 have both been derived from those observations.

To make the cosmological equations[6] as readable as possible, we have decided to use the symbol Y = TH = c/RH for Hubble time. This gets rid of one extra level of subscript. We also use "stretch factor" (S) in place of the usual redshift z, where S = z + 1, a factor that crops up all over the show. Stretch is simply the factor by which wavelengths have increased from the time that light has left an observed source, e.g. a stretch S=2 means wavelengths have been doubled by the cosmic expansion while the photons were in transit.

The User Interface and discussion below is out of date. I have updated the link to the latest Lightcone 7 calculator below, but section this still needs to be updated as time allows.

For the main calculator inputs, we simply have to tell it the values of Ynow, Yinf and S_eq. We may then also specify the output in tabular form in terms of S_upper, S_lower and the output steps in-between, either as step size, or as the number of steps (see the info-tooltip of the live user interface).

To make a connection back to the "old ways" of specifying inputs for cosmo-calculators, we also compute the conventional values and show them at the top-right. The page is pre-populated with default values and there are ample info-tooltips to make it (hopefully) easy to use. So, without further delay, please give it a try and tell us whether you think it is cool, whether it sucks, or anything in-between. It is a work in process, so you may actually still influence the tool.

Some further usage tips will follow...

Click here and take a cosmic dive: TabCosmo6.

Usage tips: see comments #3, #5 below.

The latest version is called LightCone 7, sporting a chart feature in addition to the tabular output.

-J

[1] http://cr4.globalspec.com/blogentry/20218/Cosmological-Calculator-Update

[2] http://www.physicsforums.com/forumdisplay.php?f=69

[3] It is presently thought that the universe is either spatially flat (or very near to flat), i.e. parallel lines 'here' are still parallel 'there', when considered on a large scale and viewed everywhere at the same cosmic time. However, due to the expansion, light rays that are sent out to be parallel, will not remain parallel over time; hence, spacetime is curved.

[4] Einstein (1915), "Die Feldgleichungen der Gravitation (The Field Equations of Gravitation)"

[5] The conventional Hubble constant is given in units Km/s/Mpc, i.e., a recession speed per distance. With years for time and light-years for distance, speed becomes dimensionless and the speed of light is 1. If we also convert Mega-parsec to billion light years, we get that the present H0 = 70.36 km/s/Mpc becomes 70.36/978 = 0.0712 Gly-1. Now if we invert that, we get RH_now= 13.9 Gly and TH_now= 13.9 Gy.

[6] Here are the simplified equations of CosmoLean, for those who can't live without them.

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#1

Re: Cool Cosmo Calculator

09/18/2012 7:28 PM

Nicely stated....Looks like you've been busy....Can you give an illustration of how this might be used to solve a problem? Just as a 'for example' type thing....

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#3
In reply to #1

Re: Cool Cosmo Calculator

09/19/2012 12:07 AM

Hi SE

A good example of the utility of CosmoLean is to find the largest distance D_then distance that any galaxy that we presently observe could have had. This would also give its maximum angular diameter distance.

Looking at the outputs with default inputs, we see:

SaTT_HubD_nowD_thenD_hor
10.0000.1000.5590.83930.8903.0894.653
9.0000.1110.6540.98329.9823.3315.069
8.0000.1250.7811.17228.9093.6145.568
7.0000.1430.9541.43027.6153.9456.179
6.0000.1671.2021.79826.0134.3356.942
5.0000.2001.5782.35423.9574.7917.919
4.0000.2502.1993.25821.1915.2989.207
3.0000.3333.3574.88517.2065.73510.948
2.0000.5005.9648.14810.8865.44313.262
1.0001.00013.75513.9000.0000.00015.622

We note that D_then reaches a max around S = 3, so we select 3.5, 2.5, 0.1 and get

S

a

T

T_Hub

D_now

D_then

D_hor

3.500 0.286 2.679 3.942 19.399 5.543 10.010
3.400 0.294 2.795 4.107 18.997 5.587 10.187
3.300 0.303 2.920 4.282 18.578 5.630 10.368
3.200 0.313 3.055 4.469 18.140 5.669 10.555
3.100 0.323 3.200 4.670 17.683 5.704 10.749
3.000 0.333 3.357 4.885 17.206 5.735 10.948
2.900 0.345 3.526 5.115 16.706 5.761 11.153
2.800 0.357 3.710 5.362 16.182 5.779 11.364
2.700 0.370 3.910 5.628 15.633 5.790 11.581
2.600 0.385 4.128 5.914 15.056 5.791 11.805
2.500 0.400 4.366 6.222 14.449 5.780 12.035

If we want more accuracy, we can zoom in more by selecting closer and closer upper and lower S. For accuracy on one or all columns, we may need to set the number of decimals larger before collapsing the form for 'copy and paste-friendliness'.

The max value of D_then is obvious from this output

S

a

T

T_Hub

D_now

D_then

D_hor

2.650 0.377 4.016 5.768 15.348 5.79170 11.692
2.648 0.378 4.021 5.774 15.336 5.79169 11.697
2.646 0.378 4.025 5.780 15.325 5.79173 11.701
2.644 0.378 4.030 5.786 15.313 5.79172 11.706
2.642 0.379 4.034 5.791 15.302 5.79170 11.710
2.640 0.379 4.038 5.797 15.290 5.79174 11.715
2.638 0.379 4.043 5.803 15.279 5.79172 11.719
2.636 0.379 4.047 5.809 15.267 5.79170 11.724
2.634 0.380 4.052 5.814 15.255 5.79168 11.728
2.632 0.380 4.056 5.820 15.244 5.79166 11.733
2.630 0.380 4.060 5.826 15.232 5.79164 11.737

D_then max = 5.79174 Gly at S = 2.64. That galaxy is now 15.29 Gly from us.

Does this help?

-J

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#6
In reply to #3

Re: Cool Cosmo Calculator

09/19/2012 10:56 AM

Yes thank you....with a quick study here

http://en.wikipedia.org/wiki/Observable_universe

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#2

Re: Cool Cosmo Calculator

09/18/2012 11:49 PM

constants: the local scale gravitational constant (G) and the large scale cosmological constant (Λ), both representing a curvature of spacetime, as applicable to their respective regimes.

Don't we still need a cause for these effects? (i.e. gravity, dark energy)

It is presently thought that the universe is either spatially flat (or vey near to flat), i.e. parallel lines 'here' are still parallel 'there', when considered on a large scale and viewed everywhere at the same cosmic time. However, due to the expansion, light rays that are sent out to be parallel, will not remain parallel over time; hence, spacetime is curved.

Nice explanation.

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#4
In reply to #2

Re: Cool Cosmo Calculator

09/19/2012 12:24 AM

Hi S, thanks.

"Don't we still need a cause for these effects? (i.e. gravity, dark energy)"

I guess that's what the quest for a quantum gravity theory is all about. Presently http://en.wikipedia.org/wiki/Loop_quantum_cosmology is my favorite.

I'm afraid it may perhaps be "turtles all the way down", because any cause might just be caused by a further effect, with a cause...

-J

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#5

Re: Cool Cosmo Calculator Usage tip

09/19/2012 5:10 AM

Apart from the tabular usage that I discussed in #3 above, the more visually pleasing results are graphs. This is not presently possible in the calculator, but copying a block of data to a spreadsheet and specifying a chart can be quite enlightening, e.g.

I have specified the S range of 0.01 to 10 with 100 steps (S_step = -100), copied, pasted and set up the Excel chart for the ranges shown. Note that s < 1 means "looking into the future", with the black time curve actually going to about 88 billion years at S = .01, as you can view in the calculator output table.[1] This calculator compresses the whole future into the left-most part, 0 < S < 1, while the past goes to the infinite right.

A large number of insights are 'hidden' in these graphs.[2] To name a few:

  1. T_now (at S=1) is about 13.7 Gy, which is almost the same as the Hubble time now (13.9 Gy); the black and blue curves crosses approximately at S=1.
  2. The green D_now curve heads for an intercept with the y-axis at D_now = -15.6 Gly. This is the same as the present value of our cosmic event horizon (D_hor), the red curve at S=1. If we send a message today, an observer at that distance will never receive it. We will also never receive a message sent from there today.
  3. D_hor is still growing today, but will eventually stabilize at 16.3 Gly, as will T_Hub, which is the same as the Hubble radius in the units we use. Hence, we can possibly observe a little more of the past than what we see today, but not all that much more.
  4. It is also easier to see the range where D_then maxes out: around S=2.6, exactly where purple D_then curve intersects the blue T_Hubble = D_Hubble curve. This is exactly where proper recession speed equalled the speed of light at that time; hence no photon that was then farther than that can ever reach us - all due to the present accelerated expansion.[3]
  5. Photons that reach us with stretch other than S=2.64 (smaller or larger) had to originate closer to us than D_then = 5.79 Gly.

It takes a bit of time to get used to, but eventually the insights are very rewarding, I like to think.

-J

[1] If you have not done so yet, please read the info-tooltips for each column of CosmoLean. They contain the definitions of the graphs.

[2] You can download the .jpg here for better resolution.

[3] Note that recession rate should not be thought of as a 'speed' in the normal sense of the word. It is the rate at which proper distance grows due to expansion, where proper distance is measured by our present distance measurement tools on a 'freeze frame' of the expanding cosmos. Nothing moves faster than c, but successive such measurements may throw up a recession rate exceeding c.

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