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POLL: How do you think of the Hubble rate?

  1. As a continuously compounded growth rate

  2. As an expansion speed-to-size ratio

  3. As the slope of the scale factor log: H = (ln a)' = a'/a

  4. As one over the corresponding e-fold time

  5. Several of these

  6. In a way entirely different from those listed

Multiple votes are allowed.
Results are only viewable after voting.
  1. Aug 9, 2015 #1

    marcus

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    I'd like to find out how other members usually think of the Hubble rate, by preference and/or by habit. I was unsuccessful in creating a poll in Cosmology (no "add a poll" button) so I'm hoping this thread can be moved over there.

    Here are the options I could think of:

    1. As a continuously compounded growth rate for distances (like a percentage interest rate)

    2. As the expansion speed-to-size ratio for large scale distances at a given moment in time

    3. As the slope of the log of the scale factor, i.e. H = (ln a)' = a'/a

    4. As one over the corresponding e-fold time

    5. Several of the above

    6. An entirely different way from those listed
     
  2. jcsd
  3. Aug 9, 2015 #2

    marcus

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    When I got around to voting myself just now, I voted for the two that are always coming to mind (mathematically equivalent--two sides of the same coin ) and was pleasantly surprised to see someone else had already voted ahead of me, and had selected the very same two options!
     
  4. Aug 9, 2015 #3

    marcus

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    The equation that best models the U expansion--Alex Friedmann 1922 derived by radically simplifying Einstein 1915 GR equation--this equation basic to all cosmology is about H.
    You could say that "H is how the universe thinks about its own expansion". There is no simpler direct handle on it. But what is H?
    How do you imagine it? What does it mean to you? How do you picture it evolving over time?

    I mean specifically what's called "universe time" or "Friedmann time". ( Time as measured by observers all over the U who are at rest with respect to the Background of ancient light--at rest relative to the space around them in other words, or to the expansion process itself.) That's the standard cosmic time the cosmic model runs on.

    H is a physical quantity, so it must have unit(s). The time-derivative of a pure number-valued function has the unit (Time)-1. It's slope is "number per unit time"---not a speed, but a numerical rate of change. One over a time quantity. Could that be what H is? And if so what is the physical significance of its reciprocal?

    Just some of the questions you might ask---maybe not the best, either. What other questions about H come to mind?
     
  5. Aug 9, 2015 #4

    e.bar.goum

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    That'd be me! I'm not sure if my thinking is overly literal, but hey.
     
  6. Aug 9, 2015 #5

    marcus

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    but hey. Let's have a larger format look at that Emmy Noether portrait
    Emmy.png
     
  7. Aug 9, 2015 #6

    e.bar.goum

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    Learning Noether's theorem in undergrad is really what made me properly fall in love with physics. One of those "holy crap" moments. Thus, the avatar.
     
  8. Aug 9, 2015 #7

    ShayanJ

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    Einstein had that moment about it too!ref
    its a really amazing theorem!(Actually two theorems)
     
  9. Aug 11, 2015 #8

    marcus

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    SIX of us have voted so far--thanks Demystifier, E.bar.goum, Ohwilleke, Shyan, and WhatisGravity!

    Here's how the votes stack up for the best way to think of the Hubble rate H(t):

    First place: SPEED TO SIZE RATIO

    Second place: H = (ln a)' = a'/a SLOPE OF THE SCALE FACTOR'S LOG

    Third place: Continuously compounded GROWTH RATE (like a percentage interest rate)

    Fourth place (one vote): Other, a completely different way.
     
  10. Aug 13, 2015 #9

    Chronos

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    Can we be sure the Hubble rate is absolute over time and space. I'm not sure how you would even go about testing that assumption, which is implicitly embedded in this question.
     
  11. Aug 13, 2015 #10

    marcus

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    Well physical laws typically involve approximation, they are patterns of regularity that we see and use to mirror nature.
    So I don't know about "absolute".

    But the Hubble rate certainly does change over time. And in the standard cosmic model it changes in a surprisingly simple way.
    The approximate rule is H(t) = coth(1.5t) where time t is measured in zeit units and the Hubble rate is measured in zeit-1 units.

    The question would be does the expansion rate change over space? It does not, in the standard LCDM cosmic model, and it does not in Friedmann equation models in general. Practically speaking, since the standard model gives such a good overall fit and has proven so useful, it seems like a good idea to treat it as constant over all space. That works.

    but I would say that what you "believe" is your own business. Physics models are to be worked with and tested. I don't think of them as matters of verbally-expressed "belief". the practical issue is what's the simplest best-fit mathematical model.

    BTW thanks for joining the poll, Chronos! Seven of us have voted so far, and it seems like the clear favorites are the two mathematically equivalent conceptualizations speed-size ratio and the logarithmic derivative a'/a

    Also BTW: don't you like the look in Emmy Noether's eyes. there is a touch of the comedic in mathematics.
    But so calm, unobtrusive, contemplative that we are mostly unaware of the gently celestial comedy of it.

    Schiller called it the "Feuerspiegel"---the fiery mirror that we hold up to nature. Or I think he did.

    I like it that E.bar.goum makes that portrait her self image.
     
    Last edited: Aug 13, 2015
  12. Aug 13, 2015 #11

    marcus

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    I found something like Schiller's 1785 original. In honor of Emmy Noether let's put it in her native language:

    Aus der Wahrheit Feuerspiegel
    Lächelt sie den Forscher an.
    Zu der Tugend steilem Hügel
    Leitet sie des Dulders Bahn.
    Auf des Glaubens Sonnenbergen
    Sieht man ihre Fahnen wehn.
    Soll das Dasein sich entbergen
    Wo im Chor die Engel stehen.

    Rough literal translation:

    From the fiery mirror of truth, she (Joy/nature...) smiles at the searcher.
    To the steep hill of virtue she leads the patient one's way.
    On the sunny mountaintops of belief, you see her banners waving.
    Reality shall reveal itself where in chorus the angels stand.
     
    Last edited: Aug 14, 2015
  13. Aug 14, 2015 #12

    marcus

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    EIGHT of us have voted so far--thanks Chronos, Demystifier, E.bar.goum, Nonlinearity, Ohwilleke, Shyan, and WhatisGravity!

    Here's how the votes stack up for the best way to think of the Hubble rate H(t):

    Tied for first place:
    SPEED TO SIZE RATIO and H = (ln a)' = a'/a SLOPE OF THE SCALE FACTOR'S LOG

    Second place:
    Continuously compounded GROWTH RATE (like a percentage interest rate)

    Third place (one vote):
    OTHER, a completely different way.
     
  14. Aug 16, 2015 #13
    Well, isn't the Hubble "parameter" a'/a the de facto standard in determining the expansion rate of the universe? And it's not the Hubble constant, right, it is time dependent? And the value is driven by the residual of the inflaton field?

    So I'd have to vote #3, "3. As the slope of the log of the scale factor, i.e. H = (ln a)' = a'/a"

    Is there any evidence that this is continuously compounding?

    Do we know how many e-folds there's been? 50-60-70? I wouldn't pick that answer..

    So I'm going with #3 :oldsmile:
     
  15. Aug 16, 2015 #14

    marcus

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    In first place we now have:
    As the slope of the scale factor log: H = (ln a)' = a'/a

    In second place:
    As an expansion speed-to-size ratio

    In third place:
    As a continuously compounded growth rate

    Thanks for voting, Chronos, Demystifier, DiracPool, E.bar.goum, MrNike, Nonlinearity, Ohwilleke, PeterDonis, Shyan, and WhatisGravity!
     
  16. Aug 17, 2015 #15

    marcus

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    This comment is so elementary that it's a bit embarrassing to be making it. An expansion speed-to-size ratio is what you multiply the SIZE of a distance by, to get its expansion SPEED, in other words the speed it is growing.

    Just in terms of units, now, what kind of quantity do you multiply a length by, to get a speed. The answer is a reciprocal time, like for example
    (10 years)-1 or (1 hour)-1.

    Multiplying a length by a reciprocal time is the same as dividing by some time quantity---so you get a length/time quantity, a speed.

    So the Hubble rate, whether or not we want to think of it that way, is a reciprocal time
    and, in fact, the current Hnow = (14.4 billion years)-1
    If you take any large distance and divide its length by 14.4 Gy you get the speed that distance is growing.
    (14.4 billion years)-1 is the present speed-to-size ratio.

    You can check that if you are familiar with the idea that the socalled Hubble radius (now 14.4 Gly) is the size of distances that are growing at the speed of light. If you apply the speed-to-size ratio to such a distance you get 14.4 Gly/14.4 Gy which boils down to one lightyear per year, or c.

    We know that our universe's Hubble rate is changing---gradually declining. But imagine we lived in a universe where it had stopped changing and was going to be constant from now on.
    H is the time derivative of ln a(t) and if H is constant that means ln a(t) = Ht + K
    so a(t) = eHteK

    Since H = (14.4 billion years)-1 how long does it take for this a(t) to grow by a factor of e = 2.718? Since we are talking about multiplicative factors we can ignore the constant eK.
    What we want is Δt such that H Δt = (14.4 billion years)-1 Δt = 1.
    So Δt = 14.4 billion years
    That is the e-fold time---the time needed for a(t) the size of distances to expand by a factor of 2.718.

    Assuming, that is, H had stopped changing and remained constant at the present level of
    (14.4 billion years)-1
     
    Last edited: Aug 17, 2015
  17. Aug 17, 2015 #16

    andrewkirk

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    That it is constant over space is part of the standard assumption of cosmology that the universe is spatially homogeneous, isn't it? Like with most assumptions, we can't be sure of it, but for the time being it seems pretty reasonable.

    Isn't the value an outcome of the Friedman equations, which - AFAICT - don't involve inflatons?
    It's funny how it's not a constant but is usually referred to as 'The Hubble Constant'. That had me totally confused, a little way back, until somebody gently explained to me that the Hubble 'Constant' isn't constant.
     
  18. Aug 17, 2015 #17

    marcus

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    Twenty-three of us have voted so far!
    Thanks everybody: Abtinnn, Andrewkirk, Bocaj, Chronos, Delta, Demystifier, DiracPool, E.bar.goum, Garth, HSmith, Jerromyjon, JimMcnamara, MrNike, Mugiwara, Nonlinearity, Ohwilleke, PAllen, PeterDonis, Rootone, Shyan, Vanhees, and WhatisGravity!
     
    Last edited: Aug 17, 2015
  19. Aug 18, 2015 #18

    marcus

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    I think that's a fair statement. The standard cosmic model is LambdaCDM and you can have LambdaCDM cosmology without ever needing an episode of inflation.
    Something else besides inflation can have given expansion its initial kick. There does not have to ever have been an "inflaton" field for standard cosmology to work and agree with observations made so far. For instance google "LambdaCDM bounce".
    http://arxiv.org/abs/1412.2914
    A ΛCDM bounce scenario
    Yi-Fu Cai, Edward Wilson-Ewing
    (Submitted on 9 Dec 2014)
    We study a contracting universe composed of cold dark matter and radiation, and with a positive cosmological constant. As is well known from standard cosmological perturbation theory, under the assumption of initial quantum vacuum fluctuations the Fourier modes of the comoving curvature perturbation that exit the (sound) Hubble radius in such a contracting universe at a time of matter-domination will be nearly scale-invariant. Furthermore, the modes that exit the (sound) Hubble radius when the effective equation of state is slightly negative due to the cosmological constant will have a slight red tilt, in agreement with observations. We assume that loop quantum cosmology captures the correct high-curvature dynamics of the space-time, and this ensures that the big-bang singularity is resolved and is replaced by a bounce. We calculate the evolution of the perturbations through the bounce and find that they remain nearly scale-invariant. We also show that the amplitude of the scalar perturbations in this cosmology depends on a combination of the sound speed of cold dark matter, the Hubble rate in the contracting branch at the time of equality of the energy densities of cold dark matter and radiation, and the curvature scale that the loop quantum cosmology bounce occurs at. Importantly, as this scenario predicts a positive running of the scalar index, observations can potentially differentiate between it and inflationary models. Finally, for a small sound speed of cold dark matter, this scenario predicts a small tensor-to-scalar ratio.
    14 pages, 8 figures,

    AFAIK even people who believe in an inflation episode don't ordinarily assume that the standard cosmic model involves a residue of the hypothetical "inflaton" field. It MIGHT, but that's a pretty speculative idea. Probably the simplest way to treat the cosmological constant Lambda is as just that: a constant.
     
    Last edited: Aug 18, 2015
  20. Aug 18, 2015 #19
    Thanks especially for post #5. And Schiller was a contemporary of one of my great-something grandfathers, Lessing; so I prefer to think of cosmic expansion like matter in rising dough, rather than quantitatively, and vote #6.
     
  21. Aug 20, 2015 #20

    marcus

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