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Introducing the Model T time unit "Lizzie": key to H_infinity

  1. May 30, 2015 #1

    marcus

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    The Ford Model T, affectionately known as the Tin Lizzie, is thought to have been the first widely affordable/accessible automobile. Production of the Model T started in 1908.

    I'm looking for a cosmo-friendly time unit that is short enough to be real and familiar to all of us, but at least an order of magnitude longer than a year. It should be a passage of time you can experience and remember so the concept of this amount of time is intuitively accessible. It doesn't have to have a name yet. Let's try an interval of around nine hundred weeks, and provisionally call it a "Lizzie" in honor of the Model T.
    The longterm distance growth rate H has been measured at one part per billion per Lizzie.
    The actual growth rate H(t) is observed to be gradually declining towards and leveling out at this longterm H rate. To make the unit more exactly cosmo-friendly, adjust it to be 903 weeks.

    Growth at this rate means that in a time interval of 903 weeks (one Lizzie) a generic distance increases by one billionth of its length.

    903 weeks (a little over 17 years) is a long time but not too long to grasp in terms of one's personal experience. I'm thinking of how someone's life might naturally fall into five chapters of about this size.
    Youngster (0-17) childhood and teen years
    Striver(17-35) college, striving for grades/job qualifications, connections, building career
    Midster (35-52) experienced, settled, family
    Senior (52-69) security/promotion based on seniority, "senior moments", retirement
    Oldster (69+) however it goes, from then on...
     
    Last edited: May 31, 2015
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  3. May 30, 2015 #2

    marcus

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    Thinking about the stages in one's life (eg. from when one starts having kids to when they are pretty much grown up and out of the house), the lives of ones' parents,or the lives of ones' friends, it seems possible to assimilate time at level of life chapters and get familiar with the idea of a 17.3 year interval (around 900 weeks) you can compare with the length of main stages and pace of events. I want to get away from the automatic tendency to think in terms of years and year-counts. Years are obviously not a natural unit to use in every case. The Universe knows nothing about years---they are special to this one planet. Other planets can have radically different length years and cycles of seasons. Maybe some people find it easier than I do to get a longer perspective on the passage of time. For them this could be a no-brainer, but I have to work on it a bit.

    Anyway what matters here is to have a solid conceptual grasp of the Universe's intrinsic distance growth rate of 1 ppb per Lizzie or one part per billion every 903 weeks. That is the unit growth rate H that I want us to be able to compare the current passing growth rate H(t) to. The current growth rate is declining and approaching 1 ppb per Lizzie as a limit. Numerically it is slightly above one and it is gradually settling down to one.

    Assuming readers are secure with that unit growth rate as a standard of comparison then we can say that the present rate of distance growth Hnow is 1.2, or more precisely 1.201... We can give it simply as a bare number without bothering with units. And at that point we can say that the Universe's distance growth rate, the so-called Hubble constant (except it isn't really constant) follows this equation:
    $$H(t) = \frac{e^{3t} + 1}{e^{3t} - 1}$$
    where tnow = 0.8,
    or 0.797... if more precision is needed.
    You can see how for positive time t this H(t) function is always above 1 but settles down towards 1 as e3t gets large. The expansion history of the U has been studied a lot and this really seems to be the track it is following! It could even be a little surprising, how regular the expansion process has been.
     
    Last edited: Jun 1, 2015
  4. May 30, 2015 #3

    marcus

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    I want to focus on this equation and on its inverse that you get when you solve for t in terms of H.
    $$H(t) = \frac{e^{3t} + 1}{e^{3t} - 1}$$
    $$t(H) = \frac{1}{3}\ln\frac{H+1}{H-1}$$
    This is the reciprocal relation between the distance expansion rate H and time t, when both are given in terms of natural H units---that is to say in time units of a billion Lizzies :smile:
    And this appears to be, to a very close approximation, how our universe actually runs.

    The whole exercise turns on the longterm expansion rate H, which was first measured in 1998, and is what I'm calling
    1 ppb per Lizzie. It turns on being able to assimilate and visualize that growth rate. If anyone can think of a better way to reify/realize it please propose it, even in jest. :smile: What I'm guessing is that we are all able to concretely picture something increasing in size by one part per billion, and we can all get a solid conceptual grasp and feeling about the passage of a span of time I'm calling Lizzie.
     
    Last edited: May 31, 2015
  5. May 30, 2015 #4

    marcus

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    The time unit here is a billion Lizzies. (When we say the present expansion age of the Universe is 0.8, that means 0.8 of a billion Lizzie intervals and you can figure out what 0.8 of that is in billions of years if you want.) The expansion rate unit here is the longterm rate H which is 1 ppb per Lizzie, and you can see that time and expansion rate are simply related by the equations in post #3:
    If you are given a time you can figure out what the expansion rate was then and
    if you are told the expansion rate was (or will be) such and such you can figure out the time when it was that,
    or when it will be that.

    In this thread we can explore the possibility of presenting basic quantitative cosmology without mentioning unfamiliar math stuff like "hyperbolic functions" and "hyperbolic cotangent". Basically all we need is the exponential growth function ex and the natural logarithm ln(x). It should be very "hands-on" you should be able to calculate everything with google calculator or a cell phone.

    Besides the time&expansion rate relation (post #3) there is just one other thing to add to an introduction like this, I think. That is the expansion history of a generic distance.
    It's customary to use the letter "a" to stand for the size, and eventually to scale the whole history so that a=1 at present so you get a standardized expansion history. Of course how the history looks will be a direct consequence of how the growth rate has been behaving--one determines the other--but we want the explicit overall picture of how a sample distance grows.

    It turns out that the cube of a, the cube of the size of a generic distance is inversely proportional to H2 - 1. $$a^3 \sim \frac{1}{H^2 - 1}$$
    So whatever distance we are studying, its size now anow, is related in this way to the expansion rate now, Hnow, which we know is 1.201...
    $$a_{now}^3 \sim \frac{1}{H_{now}^2 - 1} = \frac{1}{1.201..^2 - 1} = \frac{1}{0.4433}$$ So we can divide a by anow to get the standard version of the history which equals 1 at present. $$(\frac{a}{a_{now}})^3 = \frac{.4433}{H^2 - 1} $$ For simplicity let's assume that step has already been performed and a has already been normalized to equal 1 at present $$a^3 = \frac{.4433}{H^2 - 1} $$ Longterm, H declines to 1, so the denominator goes to zero, and distances increase indefinitely. We can plot the growth of a generic distance as a curve, using this formula: $$a = ( \frac{.4433}{H^2 - 1})^{1/3} $$
    To see the curve, click on the following link:
    View attachment 84208

    A useful number s to have handy is s = 1/a, the reciprocal of a. We can think of it as the "ours versus theirs" ratio for distances and wavelengths. Or the "now versus then" ratio. Say we detect the distinctive light of hot hydrogen in another galaxy, but with enlarged wavelengths. We measure the wavelengths and can tell by what factor the waves are enlarged compared with when they were emitted. If s=2 then the wavelengths now are twice what they were when emitted. And the size of a generic distance has doubled too, while the light was in transit. $$s^3 = \frac{H^2 - 1} {.4433}$$ This gives us a way to tell what the expansion rate H was when the light was emitted, and therefore from that to deduce what time the light started on its way to us.
     
    Last edited: May 31, 2015
  6. May 31, 2015 #5

    marcus

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    For people who relate well to tables of numbers, here's a sample history of the universe from a time in the past when distances were 1/50 what they are today, out to a time in the future when they will be 20 times what they are today. So the table covers a THOUSAND-FOLD expansion. And its various columns are mostly things we discussed in the past 3 posts. Our equations inter-relate these things such as time t, expansion rate H, normalized scale factor a, and its reciprocal s = 1/a.
    The new thing in the table is the distance back then to a bit of light that is going to arrive today, or the distance from us of a bit of light that we send today, at some time in the future. In either case, past or future, it is the distance then not the distance now (but multiplying by the s factor in that row would get the corresponding distance now.)
    A zeon is a billion Lizzies, and a light-zeon is a billion light-Lizzies. I'm tempted to call a light-Lizzie (i.e. 17.3 lightyears) a Beth, but I'll restrain myself. In that case a light-zeon would be a billion Beths, and the speed of light would be 1 beth per lizzie. That actually seems about right to me but I won't insist :smile:
    [tex]{\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (zeon)&D_{then}(lzeon)&H(zeon^{-1}) \\ \hline 0.020&50.000&0.003&0.046&237.099\\ \hline 0.026&37.811&0.004&0.059&155.644\\ \hline 0.035&28.593&0.006&0.076&102.218\\ \hline 0.046&21.622&0.010&0.096&67.156\\ \hline 0.061&16.351&0.015&0.121&44.135\\ \hline 0.081&12.365&0.023&0.151&29.016\\ \hline 0.107&9.351&0.035&0.186&19.088\\ \hline 0.141&7.071&0.053&0.225&12.571\\ \hline 0.187&5.347&0.081&0.267&8.299\\ \hline 0.247&4.044&0.122&0.305&5.508\\ \hline 0.327&3.058&0.185&0.332&3.699\\ \hline 0.432&2.312&0.276&0.334&2.546\\ \hline 0.572&1.749&0.407&0.292&1.836\\ \hline 0.756&1.322&0.581&0.187&1.423\\ \hline 1.000&1.000&0.797&0.000&1.201\\ \hline 1.313&0.762&1.036&0.275&1.094\\ \hline 1.724&0.580&1.292&0.655&1.042\\ \hline 2.264&0.442&1.556&1.164&1.019\\ \hline 2.972&0.336&1.825&1.837&1.008\\ \hline 3.903&0.256&2.096&2.724&1.004\\ \hline 5.125&0.195&2.368&3.888&1.002\\ \hline 6.729&0.149&2.640&5.418&1.001\\ \hline 8.835&0.113&2.912&7.427&1.000\\ \hline 11.601&0.086&3.184&10.065&1.000\\ \hline 15.232&0.066&3.456&13.529&1.000\\ \hline 20.000&0.050&3.729&18.077&1.000\\ \hline \end{array}}[/tex]

    So you can see that around time 3 million Lizzies the expansion rate was 237 ppb per Lizzie.
    And at that time (when H was 237 ppb per Lizzie) a photon could have started towards us from a distance of 46 million light lizzies and be destined to eventually get here.

    But by time 4 million it would not have gotten closer but in fact it would have been farther from us! Namely 59 million light lizzies. Just a few things to practice getting out of a table of numbers.

    You can see how far back in history the expansion rate H was much larger, how it dives down, how it is only 1.201 ppb per lizzie at present (t = 0.797) and how it levels out at one unit.

    How that photon that started towards us back around time 3 million, from a distance of 46 million, and was at first dragged back by the expansion of distance, eventually got to a maximum distance from us (when distances were roughly about 40% of their present size) and by then the expansion rate had moderated enough for it to start making progress and eventually get here.

    Jorrie set it up so that we can plot curves for these things as well as making tables, but it doesn't hurt to have some experience reading cosmic history tables as well, so I put that in.
     
    Last edited: Jun 1, 2015
  7. Jun 2, 2015 #6

    Jorrie

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    Marcus, I'm a little concerned by the unfamiliar 'units' that you propose. Even the 'zeon' did not get too warm a reception, for what I can tell, but it at least have a distinct advantage in that it allows many of the usual parameters of the LCDM model to have values around unity at present. This makes them much simpler to graph.

    But why not use nano-zeon for your 'Lizzie"? At least that's standard practice and once one can make the shift to one new unit, the rest are more 'natural'.
     
  8. Jun 2, 2015 #7

    marcus

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    Great suggestion! Very helpful, Jorrie. I'll adopt your idea. I do think that 17.3 yrs is an imaginable amount of time which one can get used to and which could gradually become familiar.
    Whereas 17.3 billion years is harder to comprehend.
    So I like picturing the important H growth rate as 1 ppb per nanozeon.
     
  9. Jun 4, 2015 #8

    marcus

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    I still suspect there may be a need for an informal nickname for the nanozeon. Start people off with a time period that can be imagined in terms of one's own and others' lives, that is part of familiar experience. I admit that "Lizzie" is an overly jocular unserious choice of nickname. I'll try to think of one or more alternative handles for this "little zeon" period of 17.3 years: a diminutive copy of zeon, that one can wrap ones' mind around and then expand a billion fold.
    Like watt and gigawatt, joule and gigajoule, bit and gigabit. Start with something intuitable and then invoke powers of ten.

    How about we say a billionth of a zeon is a zeit, for short?

    zeit to rhyme with height
    and hope that a newcomer to the subject can become familiar with the idea of a 17.3 year period in one's life as a "zeit" (an age, chapter, ...)
    then a zeon (which is the interval the universe really seems to know about) arises as a "gigazeit"
     
    Last edited: Jun 4, 2015
  10. Jun 4, 2015 #9

    marcus

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    So then the introduction goes something like this:
    Everybody knows the word "zeitgeist" for the spirit of the age or the spirit of the times we live in. We're going to take a period which is a little over 17 years or 900 weeks and call it a "zeit" , to help understand the universe expansion process.

    The fractional expansion rate of distance is gradually declining and seems to be on track to level out at a growth rate of
    1 ppb per zeit.
    At this rate, in one zeit a typical cosmic distance will increase by one billionth of its length.
    We denote this longterm growth rate H.

    Then something quite remarkable happens. It turns out that if we use this H as our scale to measure the Hubble rate H(t), the distance growth rate as it varies over time, the past history of H(t) (and the projected future history of H(t) as well!) run according to a simple formula:
    $$H(t) = \frac{e^{3t} + 1}{e^{3t} - 1}$$
    where time t is measured on a scale of billions of zeits. We'll call our unit of cosmic time (17.3 billion years) a zeon. On that scale, the present age of expansion is 0.8 zeon. (Or if you prefer to think in terms of zeits, the present-day age is 0.8 billion of that smaller time period.)

    That's I guess how a short introduction could go.
     
    Last edited: Jun 4, 2015
  11. Jun 4, 2015 #10

    marcus

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    So suppose you want to find out when was the growth rate twice H in the past history of the expansion. That, find a time t for which H(t) is 2 ppb per zeit.
    We know in the longterm distance growth rate will settle down so that in a period of one zeit (17.3 years) a distance will grow by a billionth of its size. But how recently in the past was it true that a typical distance would have grown by two billionths? We have to solve:
    $$2 = \frac{e^{3t} + 1}{e^{3t} - 1}$$
    Clearly that will be true when e3t = 3, because then the fraction is 4/2
    so t = (ln 3)/3 = 0.3662
    That is .3662 zeon, or .3662 billion zeit.
    If you want to know when it was in billions of years you can multiply by 17.3
    If we want to solve
    $$3 = \frac{e^{3t} + 1}{e^{3t} - 1}$$
    it's going to be the logarithm of 2, divided by 3, which comes out t = 0.2310.
    The larger expansion rate H we pick, the further back in time we have to go to find the time when that was the universe's distance growth rate.
    For a general H, to solve for t
    $$H = \frac{e^{3t} + 1}{e^{3t} - 1}$$
    you take the logarithm of ##\frac{H+1}{H-1}## and divide by 3.
     
    Last edited: Jun 4, 2015
  12. Jun 4, 2015 #11

    marcus

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    The formula for H(t) also tells us what the expansion rate is today. "Today" means t = 0.8 zeon, or 0.8 billion zeits.
    If more precision is needed we can use 0.797, but ordinarily 0.8 is accurate enough.
    That's how long the universe has been expanding, as of now.
    If you plug that t into the formula
    $$H(t) = \frac{e^{3t} + 1}{e^{3t} - 1}$$
    you get Hnow = 1.2 ppb per zeit, which is right. In fact the current Hubble rate is 1.2, or if more precision is needed, 1.201.
    The point of the example is that we are using the longterm growth rate H as a unit.
    The current rate Hnow is 1.2H, still 20% larger than the eventual longterm rate that it is settling down to, over time.

    How does "zeit" seem to be working out, as a nickname for nanozeon, the miniature copy of a zeon?
    Would you prefer to go back to nanozeon, or does another alternate name occur to you?
     
    Last edited: Jun 4, 2015
  13. Jun 4, 2015 #12

    Jorrie

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    'Zeit' is actually a very cute choice, because it is German for 'time'; pity we did not originally hit upon it instead of 'zeon'...

    My conservative choice would be to do away with zeon and replace it with 'zeit' and then use 'nanozeit' (or simply nz) for the 17.3 years period (only one new name).
    But, since we have already written quite a lot (on various Forums/Blogs) using 'zeon', maybe it is fine to use the two words.
     
  14. Jun 5, 2015 #13

    marcus

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    Jorrie, thanks for patience and forbearance vis à vis the tentativeness, clutter and duplication. Your mentioning blogs led me to find the 12 May blog entry where you gave a presentation of the cosmic model using H units. I hadn't seen it earlier because I was looking only for discussion board posts. It's clear and succinct. I'd like to copy it in the AeonZeon thread. It would fit into one medium sized post, I think. Let me know if that would be undesirable for any reason.
     
  15. Jun 6, 2015 #14

    marcus

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    I just realized I should be more careful in referring to the cosmological constant. Fundamentally Λ is a spacetime curvature, and we are used to quantifying curvature (in the familiar 3d space context) as reciprocal area---the metric unit is meter-2. I guess that should be the principal or default way to talk about it, but, among other ways, people also refer to Λ as a reciprocal square time---e.g. in second-2 terms---as the Wikipedia article notes:
    http://en.wikipedia.org/wiki/Cosmological_constant#Positive_value
    ==quote==
    ...There are other possible causes of an accelerating universe, such as quintessence, but the cosmological constant is in most respects the simplest solution. Thus, the current standard model of cosmology, the Lambda-CDM model, includes the cosmological constant, which is measured to be on the order of 10−52 m−2, in metric units. Multiplied by other constants that appear in the equations, it is often expressed as 10−52 m−2, 10−35 s−2, ...
    ==endquote==
    By a nice coincidence the current estimate of Λ corresponds rather closely to the latter. I tend to think of Λ as a curvature which is most conveniently expressed as reciprocal time squared (in spacetime geometry the distinction between time and distance is often blurred, people set c = 1, a curvature might be the reciprocal of either one squared.) But this is potentially confusing! If the standard way to quantify Λ is reciprocal area then the "per square time" version is c2Λ. Multiplying by c2 takes care of the units by changing meters to seconds in the denominator.

    So the Friedmann equation with Λ should say:
    $$H^2 - \frac{\Lambda c^2}{3} = \frac{8\pi G}{3 c^2} \rho$$
    And instead, I've often, if not always, been leaving the c2 off and just writing Λ.
     
    Last edited: Jun 6, 2015
  16. Jun 6, 2015 #15

    marcus

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    The nice coincidence is that if you type "(17.3 billion years)^-2" into google, you get back
    (17.3 billion years)^(-2) = 3.35519999 × 10-36 s-2
    so that (when Λ is multiplied by an invisible c2 :smile:) the current best estimate for the cosmological constant is
    Λ = 3H2 = 3(17.3 billion years)-2 = 1.00656..× 10-35 s-2 ≈ 1.007 × 10-35 s-2
    so the actual quantity turns out, by accident, to be really close to the casual order-of-magnitude estimate.
     
    Last edited: Jun 6, 2015
  17. Jun 6, 2015 #16

    marcus

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    To make a clearer connection to, for instance, Wikipedia on the Friedmann equation:
    http://en.wikipedia.org/wiki/Friedmann_equations
    http://en.wikipedia.org/wiki/Friedmann_equations#Equations
    ==quote==
    The first is:
    050472714ebcc00ada8b6d64b5cc4a04.png
    which is derived from the 00 component of Einstein's field equations.
    ==endquote==
    And there ρ is a mass density, not an energy density. In the flat or nearly flat case where spatial curvature is negligible, we take k=0, so the lefthand side is just H2.
    then we move Λc2/3 over to the left side.
    If ρ is to be an energy density then the denominator on the right must be 3c2 instead of 3.
    $$H^2 - \frac{\Lambda c^2}{3} = \frac{8\pi G}{3 c^2} \rho$$
    In either case the righthand side works out to be the square of a growth rate--in metric terms it would be a second-2 quantity.
    So unitwise it matches the lefthand side.
     
  18. Jun 7, 2015 #17

    Jorrie

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    I think the c2 in the RHS denominator mixes the units up, because the RHS then comes out as meter-2, not so? It is sometimes good to re-institute c everywhere, but one must be careful...

    BTW, if you want to copy my humble Engineering Blog post here, you are welcome. Edit as you think necessary.
     
  19. Jun 7, 2015 #18

    marcus

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    Thanks, Jorrie. BTW correct me if I'm wrong--isn't it true that ρ gets used both for mass density and energy density, so one has to specify which is meant? I've tried to make clear that I'm treating the density of matter in terms of its energy equivalent.

    This could be considered a bit roundabout. Since we are mainly dealing with non-relativistic matter where the natural density measure is mass density.
    If ρ were a mass density then Gρ would have units (time)-2.

    But I think we've been consistently treating density (of matter and radiation combined, although matter dominates mostly) in terms of the energy equivalent. I'd be reluctant to change. Let me know if you aren't comfortable with ρ as energy density.

    With ρ as energy density we need the c2 in the denominator. Gρ/c2 has units time-2
     
  20. Jun 7, 2015 #19

    Jorrie

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    My problem stems from the fact that cosmic critical density is usually given as as ρcrit ~ 10-26 kg/m3, not J/m3. My suspicion is that using ρ (without the c2) as an energy density may cause some confusion - I think physicists normally use u for energy density, but I may be wrong.

    I suppose it does not really matter if the use-convention is specifically stated.

    PS: we often speak about the energy density parameters in some forms of the Friedman equations, but they are dimensionless ratios, so units cancel.
     
  21. Jun 7, 2015 #20

    marcus

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    Well we can try out a compact presentation treating ρ as a mass density (dropping the c2 from the denominator) and since everybody knows Wikipedia and has immediate access to it, how would it be to start with a reference to their form of the Friedmann equation?

    That in turn links to their article on the Einstein Field equation so anyone who wants to get some background can go there.

    BTW can you edit your engineering Blog post, if you want to make changes (like dropping the c2 and making it mass density)?
     
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