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Embarrasingly Basic Clarification

  1. Jun 28, 2015 #1
    This is an embarrassing clarification to request, but I keep getting confused, even when sort of following the hypersine model thread and others like it.

    Is this correct?

    [itex]a(t)[/itex] "Scale Factor" - length multiplicative factor, a function of time as experienced by an observer co-moving with the CMB
    [itex]\dot { a } (t)\quad =\quad \frac { { d }^{ ' } }{ dt } a(t)[/itex] "rate of change of length" or "rate of expansion"
    [itex]\ddot { a } (t)\quad =\quad \frac { { d }^{ '' } }{ dt } a(t)[/itex] Rate of change of "rate of expansion", aka [itex]\lambda [/itex], aka the "cosmological constant".

    When someone says, in the context of cosmology that "space-time" looks flat they mean that [itex]\lambda [/itex] is very close to one, that the rate of expansion or the rate of change in length seems nearly constant.

    When someone talks about positive cosmological curvature they mean [itex]\lambda [/itex] is positive and the rate of expansion is increasing. This is associated with [itex]\Omega [/itex].

    too many rates of, then giving them new names.... oof
  2. jcsd
  3. Jun 28, 2015 #2


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    No, this is not the same thing as the cosmological constant. ##\ddot a## would be non-zero (but negative) if there was no cosmological constant.

    Neither. And nobody will say that space-time is flat since it is not. However, space may be flat for any given cosmological time. This is related to the energy content of the Universe and not to the acceleration or the cosmological constant.
  4. Jun 28, 2015 #3
    Second paragraph - is where I got confused

  5. Jun 28, 2015 #4


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    ##\Omega## describes the total energy content of the Universe. As per my previous post, it is therefore related to whether or not space is flat, open, or closed. Energy content may be in the form of matter, radiation, dark energy, or anything else which happens to be in the Universe.
  6. Jun 28, 2015 #5
    I think I knew that part. Sorry, third paragraph. The sentence that gave me trouble startsat "As far as astronomers can see space is roughly flat...no one has detected non-trivial spacetime curvature at the scale of the universe"

    when you say "space is flat, open or closed" do you mean space-time? I can't help picturing Big Crunch etc over time?

    Also, isn't correct to understand the cosmological constant as being relevant, in terms of effect, in terms of rate of change of expansion, even if it's not equivalent. In other words I am not completely misunderstanding it, or d2/dt of a(t).
    Last edited: Jun 28, 2015
  7. Jun 28, 2015 #6


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    This isn't the notation that is usually used.

    First, the "rate of expansion" is defined as ##H = \dot{a} / a##. This is a fractional rate of change of distances between objects.

    The physics links the matter in the universe to the expansion rate through the Friedmann equations, the first of which can be written as:

    [tex]H^2 = {8 \pi G \over 3} \rho - {k \over a^2}[/tex]

    Here ##\rho## is the total energy density of the universe (including mass energy) at a given time, and ##k## is the spatial curvature.

    What you see here is a balance between the rate of expansion ##H## and the energy density of the universe ##\rho##. To simplify things, it's easiest if we just consider ##\rho## being made up of matter, and ignore dark energy. In that situation, if ##{8 \pi G \over 3} \rho## is bigger than ##H^2##, then the gravitational attraction of the matter will cause the expansion of the universe to eventually stop, and the universe will recollapse on itself. If the expansion rate is much faster than the matter density in these units, then it will expand forever.

    This is exactly analogous to the situation of throwing a ball in the air near the Earth. If you throw it as hard as a human can throw it, it won't be moving fast enough to leave the Earth for good: it will fall right back down in just a short time. If you had super strength, though, you might be able to throw it so hard that it leaves the Earth entirely, never to return. The initial conditions of the universe, just like the initial velocity of the ball when it leaves your hand, determines the ultimate path of the universe into the future.

    The reason why it works this way is that as the universe expands, the matter density falls as ##1/a^3##. This comes from the increasing volume of the universe after expansion. So as the expansion continues, the matter density gets lower and lower faster than the impact of curvature on the expansion. If ##k## is positive (which indicates a universe that started out expanding slower), then eventually the ##\rho## term will shrink enough that the ##k## term dominates, causing the expansion to reverse.

    But a cosmological constant doesn't work that way. In the above equation, you can model the cosmological constant as an energy density that does not change as the universe expands. In that situation, all of the matter and the impact of the curvature dilute away, becoming much smaller than the cosmological constant. Eventually there is nothing left of the energy density of the universe but the cosmological constant, and the expansion rate is precisely given by that constant.

    Does that help?
  8. Jun 28, 2015 #7
    Very much, thanks Chalnoth.

    Gotta study it some.
  9. Jun 29, 2015 #8
    Yep, the Friedman equation(s), their structure, derivation, history are really what I have been missing when trying to follow some of the threads here. I was latched onto the FLRW eq. Didn't get that it was an input to the more complete model (is that missing it still?)

    I get the idea of omega, energy density, k, Lambda and the freeze vs. crunch scenario generally, but for instance that nuanced difference ascribing some of the expansion, or rate of change of expansion to energy density, and then this other term Lambda that was sort of lurking as a weak constant in there, definitely right on my confusion.

    I want to understand why it can't be that there is one overall expansive DOF, rather than one (relativistic energy density) + the weird residual lambda. I believe it but I want to get a feel for it, which is why I wish I could follow Marcus' threads more intuitively.

    Thanks again.
  10. Jun 29, 2015 #9


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    I don't know what you mean by "one overall expansive DOF".

    I think the best way to understand the Friedmann equation may be to realize that it's all about space-time curvature. The space-time curvature is always going to be a function of how much stuff there is in the universe. You can rewrite the Friedmann equations, for example, like so:

    [tex]H^2 + {k \over a^2} = {8 \pi G \over 3} \rho[/tex]

    Here all the curvature terms are on the left, while the energy density is on the right. The total space-time curvature is ##H^2 + k/a^2##. The reason that there are two terms is that the FLRW metric splits the curvature between a space-only component and a component that mixes space and time. The component that mixes space and time manifests itself in the expansion of the universe.
  11. Jun 29, 2015 #10
    Yeah, that is it exactly.... thanks.
    A spare and dense equation that does exactly help clarify exactly where I've got an inkblot problem. It keeps trying to switch back and forth between the idea that energy density is a causal function of the curvature rather than the curvature being caused by the stuff. I know that is pretty goofy. That does seem to really capture it though.

    where is Lambda in that one? I assume it would slot to the left with all the other "curvature" but then... it is considered an energy density right? :confused:

    The thing that is bugging me, just this minute, as I try to think of why in the world I can't just see the energy density first, easily.

    If QM Superselection, the process through which objects in space-time, get built, exist, move etc, is space-like non-local, that just seems to lay a confusing crack in that notion that the field is derived from the object and not the other way around.

    In the absence of anything at all, does an object occur first or an instantaneous field... distortion, or symmetry?I can imagine what seems to me like a puzzle of simultaneity is easily perceived in all the appropriate glory of intuitive causal flow by someone skilled at the formalism. Or maybe I'm just getting a little better handle on what everyone finds so interesting...
    Last edited: Jun 29, 2015
  12. Jun 29, 2015 #11


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    The thing is: it's both. The amount of stuff constrains the curvature, while the curvature impacts how the stuff moves and changes over time (and to make things even more complicated, even the motion of the stuff impacts the curvature). The two are interrelated and cannot be separated from one another.

    You can put it on either side, depending upon how you want to think about it. You can think of it as just an extra term in the curvature that is placed on the left, or as a constant energy density that is placed on the right. They're mathematically identical ways of describing the same thing.
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