Computing the Age of the Universe with Mathcad

[DanMarotta]

Mathscinotes, the author of Math Encounters blog produces some great pieces. Here's his latest post, Computing the Age of the Universe.

Do you find these resources helpful or educational?

Roger Mansfield, I feel like this blog post would be right up your alley?

-Dan

[RogerMansfield]

Yes, they are both helpful and educational. The graph of recessional velocity vs. distance, nicely done in Mathscinotes, was first presented to the astronomical community by Edwin Hubble in 1929.

Simon Singh reproduces Hubble's graphs and comments as follows [1, pp. 257-258]:

"So, according to Hubble and Humason's observations, all the matter in the universe was concentrated into a relatively small region roughly 1.8 billion years ago and has been expanding outward ever since. The picture completely contradicted the established view of an eternal unchanging universe. It reinforced the notion put forward by Lemaitre and Friedmann that the universe began with a Big Bang."

Cosmological theory predicts that if the universe originated in a Big Bang, then there should now (in modern times) be a remanent cosmic microwave background (CMB*) radiation, detectable in all directions of space, consistent with the radiation emitted by a black body at 2.7 degrees Kelvin. This is what John C. Mather reported in 1990 [2, pp. 234-235].

The currently accepted value for the age of the universe is about 13.7 billion years since Big Bang. But Mathscinotes calculates 13.5 billion years and Hubble obtained 1.8 billion years. Why the differences, when all three sources are using the same plot of recessional velocity (Doppler redshift) vs. distance?

The answer is simply explained via Mathcad. When one fits a straight line to the curve using Mathcad's "line" function, the addition of more data, or refinement of the existing data, will usually shift the slope and y-intercept of the line a bit. Hubble had very little data to work with in 1929, but now in 2011 we have much more data to work with.

*I did a Mathcad worksheet on Dr. Mather's CMB results a few years ago. It is attached as a Mathcad 11 file and as a pdf file. It was the twelfth worksheet in the "Mathcad Worksheets by Astroger" suite, about which you can find more at http://mathcadwork.astroger.com.

[1] Singh, Simon. Big Bang: The Origin of the Universe (Fourth Estate, HarperCollins/Publishers, 2004).

[2] Mather, John C. and Boslough, John. The Very First Light (BasicBooks, HarperCollins/Publishers, 1996). -- I had the good fortune of several conversations with Dr. Mather at the University of Denver (DU) in 2005, just months before he won the Nobel Prize. My friend Dr. Bob Stencel, William Herschel Womble Professor of Astronomy & Physics at DU, hosted Dr. Mather's visit to the DU campus.

[RogerMansfield · Accepted solution]

In my PlanetPTC post of February 3, I did not mean to shift the discussion away from Edwin Hubble and toward John Mather. It is just that the age and temperature of the universe are intimately related.

If Edwin Hubble can be said to have determined the age of the universe by means of his galactic recessional velocity vs. distance graph, then John Mather and his COBE colleagues took the temperature of the universe.

Because what John Mather and the COBE team showed is that the universe, as it exists today, behaves exactly like a black body radiating at 2.725 degrees Kelvin.

Given that it took the universe 13.7 billion years to cool down to its present temperature of 2.725 degrees Kelvin, as measured by the COBE team, how long will it take for the universe to cool down by one more degree, to 1.725 degrees Kelvin?

Using Hubble's and Mather's results, we can answer this question, at least to a first approximation. And we do not have to use any high-powered math or physics to come up with a reasonable answer.

So how long do you think it will take for the universe to cool down by one more Kelvin degree? Choose one:

A. A year.
B. A hundred years.
C. A thousand years.
D. A million years.
E. Much longer.

Or can you calculate an answer? (Hint: see the graph below.)

To see what I have calculated, download the attached Mathcad 15 worksheet or its rendering in Adobe PDF format.

[DanMarotta]

If the universe has taught us anything, it's that everything takes much longer. I'm going with E. Much longer

[StuartBruff]

RogerMansfield wrote:

In my PlanetPTC post of February 3, I did not mean to shift the discussion away from Edwin Hubble and toward John Mather. It is just that the age and temperature of the universe are intimately related.

If Edwin Hubble can be said to have determined the age of the universe by means of his galactic recessional velocity vs. distance graph, then John Mather and his COBE colleagues took the temperature of the universe.

Because what John Mather and the COBE team showed is that the universe, as it exists today, behaves exactly like a black body radiating at 2.725 degrees Kelvin.

Given that it took the universe 13.7 billion years to cool down to its present temperature of 2.725 degrees Kelvin, as measured by the COBE team, how long will it take for the universe to cool down by one more degree, to 1.725 degrees Kelvin?

Using Hubble's and Mather's results, we can answer this question, at least to a first approximation. And we do not have to use any high-powered math or physics to come up with a reasonable answer.

So how long do you think it will take for the universe to cool down by one more Kelvin degree? Choose one:

A. A year.
B. A hundred years.
C. A thousand years.
D. A million years.
E. Much longer.

Or can you calculate an answer? (Hint: see the graph below.)

To see what I have calculated, download the attached Mathcad 15 worksheet or its rendering in Adobe PDF format.

OK. It's late, I haven't got M15, I'm tired, my eyes aren't what they used to be, my brain never was and I'm not wearing sun-glasses. But ... if the Earth is approx 0.45*10^10 yrs old, then the temperature ought to have been around 4000 K at creation - seems a little high. It tends to imply that the the furthest galaxies (around 12 to13 billion light years) would have to have been incredibly hot for the redshift to leave them visible?

Stuart

[RogerMansfield]

Stuart,

You must be really tired, because you failed to see that I attached an Adobe PDF file precisely for folks who either don't have Mathcad 15 or don't have Mathcad at all -- Mathematica and Matlab visitors are welcome.

You simply have got to read the Blackbody_Universe worksheet to see my assumptions and how I obtained my numbers.

This thread that Dan started is intended to be entertaining as well as thought-provoking and informative. Please take the "quiz" and then find my own answer in the worksheet. See if my analysis makes sense to you. Point out where you think the reasoning might be weak or invalid, but please, only after you have read the worksheet.

I didn't circulate this around for peer review before I posted -- after all, this is a forum, not a journal. If I made any mistakes, I'd like to know, and I'll correct them.

Roger

P.S. The .xmcd format should open in Mathcad 14, too. I think I could accommodate a request for the worksheet in any Mathcad version going all the way back to Mathcad PLUS 6 for Macintosh. But since I have provided a PDF file, that should not be necessary.

[StuartBruff]

RogerMansfield wrote:

Stuart,

You must be really tired, because you failed to see that I attached an Adobe PDF file precisely for folks who either don't have Mathcad 15 or don't have Mathcad at all -- Mathematica and Matlab visitors are welcome.

Yep, I really was tired, Roger. I noticed the attachments, but was having enough difficulty keeping the eyelids open even with matchsticks, let alone facing the prospect of actually thinking. 🙂

However, I was sufficiently less unalert to notice your hint to look at the graph and that's what I did. I do, however, accept there is a very high likelihood that I may have misread your graph.

You simply have got to read the Blackbody_Universe worksheet to see my assumptions and how I obtained my numbers.

This thread that Dan started is intended to be entertaining as well as thought-provoking and informative. Please take the "quiz" and then find my own answer in the worksheet. See if my analysis makes sense to you. Point out where you think the reasoning might be weak or invalid, but please, only after you have read the worksheet.

I didn't circulate this around for peer review before I posted -- after all, this is a forum, not a journal. If I made any mistakes, I'd like to know, and I'll correct them.

I think this forum counts as a peer review mechanism. 🙂 I won't necessarily have the ability to spot any mistakes in your argument, but I still think that figure for the temperature 4.5 billion years ago looks way too high.... in fact, let's have a quick Google ... Wiki indicates the temperature to be around 3000 K at ~400,000 yrs, which allowed recombination to take place. What do you calculate the temperature to have been?

P.S. The .xmcd format should open in Mathcad 14, too. I think I could accommodate a request for the worksheet in any Mathcad version going all the way back to Mathcad PLUS 6 for Macintosh. But since I have provided a PDF file, that should not be necessary.

I've got M14 and I can see the process running but I can't see actual GUI. It was fine until I removed the test version of M15 I had (limited time licence) and now I can't see anything despite removing all versions of Mathcad and re-installing M11 and M14. M11 appears to work.

[RichardJ]

I've got M14 and I can see the process running but I can't see actual GUI. It was fine until I removed the test version of M15 I had (limited time licence) and now I can't see anything despite removing all versions of Mathcad and re-installing M11 and M14. M11 appears to work.

Did you also go through the registry and delete all "Mathcad" keys after uninstalling? IIRC this fixred a problem for me a few years ago.

[StuartBruff]

Richard Jackson wrote:

I've got M14 and I can see the process running but I can't see actual GUI. It was fine until I removed the test version of M15 I had (limited time licence) and now I can't see anything despite removing all versions of Mathcad and re-installing M11 and M14. M11 appears to work.

Did you also go through the registry and delete all "Mathcad" keys after uninstalling? IIRC this fixred a problem for me a few years ago.

No, I didn't. I can see much joy ahead of me ... 😞

I also had this sudden feeling of deja vu and thought the word 'zap' should be somewhere in your reply.

Was Jean sentenced to exile for life or can he apply for parole?

Stuart

[RichardJ]

No, I didn't. I can see much joy ahead of me ... 😞

I'm not sure about joy, but you might have plenty of excitement ahead of you. Probably the easiest way to get an adrenaline rush is to mess with the registry and then reboot to see what happens. If you want real excitement, do that without either backing up the registry or creating a restore point first

I also had this sudden feeling of deja vu and thought the word 'zap' should be somewhere in your reply.

Was Jean sentenced to exile for life or can he apply for parole?

I don't know, but I suspect it was for life. A few people got banned from the Collab in the past, and I don't think any of them ever came back.

Thinking of Jean though, maybe your problems with 14 are due to "The Malicious Bug"! It's all part of a PTC plot to take over the world! I can already see the army of black-clad Ribbons marching out of Needham to subjugate anyone that doesn't own a copy of Prime!

[StuartBruff]

Richard Jackson wrote:

No, I didn't. I can see much joy ahead of me ... 😞

I'm not sure about joy, but you might have plenty of excitement ahead of you. Probably the easiest way to get an adrenaline rush is to mess with the registry and then reboot to see what happens. If you want real excitement, do that without either backing up the registry or creating a restore point first

Hey! I hadn't thought of that! Thanks, I'll give it a go.

Was Jean sentenced to exile for life or can he apply for parole?

I don't know, but I suspect it was for life. A few people got banned from the Collab in the past, and I don't think any of them ever came back.

That sounds ominous. Even murderers and Athenian exiles can get away with less than that.

Thinking of Jean though, maybe your problems with 14 are due to "The Malicious Bug"! It's all part of a PTC plot to take over the world! I can already see the army of black-clad Ribbons marching out of Needham to subjugate anyone that doesn't own a copy of Prime!

It's all right. I can deal with them. The rate of change of momentum wrt time can have a strong influence on the weak-minded.

Primetrooper: Let me see your Mathcad.

Me: [with a small wave of my hand] You don't need to see my Mathcad.
Primetrooper: We don't need to see your Mathcad.
Me: These aren't the version numbers you're looking for.
Primetrooper: These aren't the version numbers we're looking for.
Me: I can go about my calculations.
Primetrooper: You can go about your calculations.
Me: Translate along.
Primetrooper: Translate along... translate along.

[DanMarotta]

It's all right. I can deal with them. The rate of change of momentum wrt time can have a strong influence on the weak-minded.

Primetrooper: Let me see your Mathcad.

Me: [with a small wave of my hand] You don't need to see my Mathcad.
Primetrooper: We don't need to see your Mathcad.
Me: These aren't the version numbers you're looking for.
Primetrooper: These aren't the version numbers we're looking for.
Me: I can go about my calculations.
Primetrooper: You can go about your calculations.
Me: Translate along.
Primetrooper: Translate along... translate along.

Brilliant!

[StuartBruff]

DanMarotta wrote:

It's all right. I can deal with them. The rate of change of momentum wrt time can have a strong influence on the weak-minded.

Primetrooper: Let me see your Mathcad.

Me: [with a small wave of my hand] You don't need to see my Mathcad.
Primetrooper: We don't need to see your Mathcad.
Me: These aren't the version numbers you're looking for.
Primetrooper: These aren't the version numbers we're looking for.
Me: I can go about my calculations.
Primetrooper: You can go about your calculations.
Me: Translate along.
Primetrooper: Translate along... translate along.

Brilliant!

[StuartBruff]

Well, this is weird. I completely uninstalled all versions of Mathcad and then went through my registry like a politician through bribes, deleting every single reference to PTC, Mathsoft, Mathcad, Prime,etc that I could lay my hands on.

I then re-installed M11 (and failed to get a licence), all of the upgrades & extensions. No problem, M11 boots up and runs a few worksheets.

Hunted high and low for my M13 disk, can't find it. Found my M7, M8, M2000, M2001i & M12 disks, but decided to give them a miss.

Next step, M14. Installation goes smoothly ... but no M14 on the screen - I can see it in Task Manager, steadily building up the MiB, but not elsewhere.

Restart PC, same result - ничего! As I happen to have the extensions, I gave a mental shrug and installed one. Result - happiness! M14 is now displaying.

???

Stuart

[RichardJ]

That is bizarre. It's hard to imagine why installing one of the extensions would fix the problem.

[AlanStevens]

Roger has assumed the universe cools via Newton's Law of Cooling to the 'nothing' outside the universe. However, 'Newton' requires the presence of a fluid to convect the heat away. 'Nothing' doesn't convect too well! Try having the heat lost by radiation (assuming perfect black-body radiation). This should also satisfy Stuart's concern about the high temperature at the time of the formation of the earth.

Alan

[RogerMansfield]

Yes, Alan, thanks. That's a better way of wording it. Any way you word it, I am obviously assuming that the rate of cooling is proportional to the Kelvin temperature.

That's the same first-order ODE that we use to describe radioactive decay (rate of decay is proportional to amount of radioactive species remaining). But I wanted to hark back to Newton, since he modeled the process of cooling via dT/dt = alpha * T many years before radioactivity was discovered.

Stuart, you are asking, what was the temperature of the universe when Earth was formed, as predicted by the decaying exponential model? That's a good question. I'll check to see what Gamow's equation predicts for that, too, and comment here. But in the meantime, you can do both checks, too, if you want, since you have the Blackbody_Universe worksheet.

Roger

Update to this post, a few hours later ...

When I compute, via the decaying exponential model, the universe's temperature at 13.7 - 4.5 = 9.2 billion years since the Big Bang, I get 4,314 degrees Kelvin. When I compute the same via Gamow's equation*, I get about 28 degrees Kelvin. (*Gamow, The Creation of the Universe, p. 40).

But if, using the decaying exponential model, I start with a temperature of To = 3000 degrees Celsius + 273 degrees = 3273 degrees Kelvin, at a recombination epoch of 13.7 - 0.38 = 13.32 billion years ago, and take T = 2.725 degrees Kelvin at present, I get a recalibrated alpha that now yields a temperature of about 30 degrees Kelvin 4.5 billion years ago.

That's only about two Kelvin degrees higher than the temperature yielded by Gamow's equation.

Stuart, should I salvage the decaying exponential model by using the temperature at recombination (3273 degrees Kelvin, 13.32 billion years ago) to recalibrate the cooling constant, or should we abandon this model?

Update to this post, a day later ...

I updated the worksheet, assuming, as stated above, that recombination occurred 380 million years after the Big Bang. The difference between the exponential decay model and Gamov's model is now about two degrees, as also stated above.

But I made an error -- recombination is thought to have occurred 380 thousand years after the Big Bang, not 380 million years. With this correction, the Kelvin temperature of the universe after 4.5 billion years now computes as 28 degrees, not 30 degrees. That's because recombination occurred so "quickly" after the Big Bang, only 380 thousand years after it, that its time since Big Bang is negligible in the calculations. The numbers that determine the cooling constant are 3273 degrees Kelvin and 13.7 billion years ago (not 13.32 billion years ago).

Agreement between the exponential decay model and Gamow's equation, to within a fraction of a degree for this particular epoch, indicates to me that the model is reasonable, and that Gamow's own calculations, made more than 61 years ago, are still worthwhile.

The calculations also support your Answer E, Dan, that it will take much longer than a million years for the universe to cool down by one more Kelvin degree.

Cosmology is fascinating and this has been fun. But I really come to PlanetPTC mostly to stay engaged with the origin and evolution of Mathcad!

Roger

P.S. When at first I found the error just noted, I posted that the model fails. But then I realized that, since recombination occurred so "soon" after the Big Bang (380 thousand years, not 380 million years), we can take the calibrating time and temperature as 3273 degrees Kelvin and 13.7 billion years ago. Somebody else please check my calculations. Stuart, I am posting the Mathcad 11 worksheet, too.

[RogerMansfield]

To Alan, Stuart, and All Others who have subscribed to this thread, My apology for the many posts on this topic that I did last Thursday evening, February 10-11, and the many e-mail notifications that you received as the result. I did not realize, until the weekend, that each and every post was being announced via an e-mail to you. As author of the posts, I myself never received any of the e-mail notifications. Dan Marotta confirmed to me that all of the posts had been e-mailed. So if you read all these posts, you can see that I found an error in my work that I thought invalidated the analysis, and so I posted that I was abandoning the exponential decay model. But then later in the evening, I redid the calculations and saw that the results got better, not worse. So I posted that. The last post, reporting success with the model, was what you see above. And that will be my final post on this topic unless any of you has any questions or comments. Roger Mansfield

[AlanStevens]

RogerMansfield wrote:

To Alan, Stuart, and All Others who have subscribed to this thread, My apology for the many posts on this topic that I did last Thursday evening, February 10-11, and the many e-mail notifications that you received as the result. I did not realize, until the weekend, that each and every post was being announced via an e-mail to you.

Not a problem for me any more - I switched off all email notifications some time ago - they were a real pain!

Alan

[RogerMansfield]

Update on the occasion of your selection as PlanetPTC Hero (October 2012):

The assumption of a decaying exponential Kelvin temperature made for a great Mathcad animation. But, Alan, as you pointed out, Newton's law of cooling could not possibly apply.

Subsequently, after visiting more of the literature of cosmology, I have found a currently accepted cosmological model, the relativistic Euclidean, or Einstein-de Sitter model (see Karttunen et al., Fundamental Astronomy, 3rd Edition, Springer, 1996, Section 20.4, "The Friedmann Models").

If I understand the model, the universe cooled from recombination to the present according to the law K = C * t^-2/3, where K is the Kelvin temperature, t is time, and the constant C can be computing using the assumed age of the universe (13.7 billion years) together with the current temperature estimate (Nobel laureate John Mather's COBE measurement of 2.725 Kelvins).

While less appealing (at least to me) than a decaying exponential, the K = C * t^-2/3 curve appears to have a much better foundation in modern cosmological theory.

The updated Mathcad 15 worksheet and video are attached.