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Particle horizon and inflation

  1. Apr 28, 2004 #1


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    To calculate the size of the observable universe, one has to calculate the current distance to the particle horizon (t0: today, c = 1):

    [tex]\int_{0}^{t_0} \ dt / a(t)[/tex] (1)

    To be able to calculate the integral one has to find an expression for a(t). With some assumptions one can take

    [tex]a(t) = (t/t_0)^\frac{2}{3}[/tex] (2)

    and get an reasonable result for the integral: 3 c t0.
    (see e.g. http://www.astro.ucla.edu/~wright/cosmology_faq.html#DN)

    Now, my question: we know that there is an inflationary period during from t = 10^-35 sec. to t = 10^-30 sec. or whatever. During this period the dependence of a(t) is different than equation (2) above and thus the integral (1) should be calculated in two steps.

    Although this period is short, it is indeed relevant because there is a huge expansion (an exponential dependence of a)

    I have never seen the particle horizon to be calculated in this way. I assume I am missing something, since, in such a case, the observable universe would be far bigger than 3 c t0. So, what is wrong?

    Last edited: Apr 28, 2004
  2. jcsd
  3. Apr 28, 2004 #2


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    this is a good puzzle
    I dont see how to resolve this. I am sure somethings wrong
    but cant say what. I hope somebody else shows up who can

    Searching for help, I found a recent set of lectures on cosmology topics.

    Lecture 4 is on inflation

    around page 9 of this lecture it has an "Idea of Inflation" section:
    -------quote from Bartelmann's lecture 4------

    • c/H is the Hubble length,
    c/(aH) is the comoving Hubble length

    • increases typically because H
    rapidly decreases as Universe

    • flatness problem would be solved if
    c/(aH) could shrink for some time...

    my thought is we are used to imagining situations where the
    H is decreasing and the hubblelength is increasing
    indeed the hubblesphere expands so rapidly that it engulfs
    light that was initially being swept away from us so that we end up
    being able to observe very rapidly receding regions of space

    but Bartelmann seems to be telling us that during inflation the
    hubblelength is DECREASING, intuitively neighboring space is receding from us at such a rapidly accelerating rate that our observable universe is shrinking

    hellfire, temporarily at least I am confused. I had swept the inflationary epoch under a kind of mental rug and was only thinking of cosmological parameters applied to after inflation had stopped. your paradox puzzles me
    also. I will try to find more stuff that bears on this

    I am used to thinking of the particle radius as the radius of the observable universe----could it be that these two things no longer match? we may get help from some of the others here at PF
  4. Apr 28, 2004 #3


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    here's another link referring to "shrinkage" during inflation


    it has some diagrams with red circles that get smaller
    dont mind saying that what seems to go on during inflation is unintuitive for me

    here's a quote from Watson:
    "...What does this mean? This implies that during a period of inflation the comoving frame SHRINKS! Remember that the comoving coordinates represent the system of coordinates that are at rest with respect to the expansion. In other words, instead of viewing the spacetime as expanding it is equally valid to view the particle horizon as shrinking..."

    Scott Watson's TOC:

    Here is a Berkeley copy of a Harvard page that links to this Watson page and has other online references about inflation:

    for completeness here's a link to Ned Wright
    Last edited: Apr 28, 2004
  5. Apr 28, 2004 #4


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    new article by Alan Guth called simply

    seems to be pedagogical, partly introductory
    partly survey of different scenarios
    might be useful
  6. Apr 29, 2004 #5


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    Yep, marcus, I also thought I had understood inflation, but I am bluffed with this problem. I think I am not able to go through all references you gave. Anyway, I know Watson's exposition and, in fact, my question was partially motivated by equations (50) and (51) of the link you gave us (http://nedwww.ipac.caltech.edu/level5/Watson/Watson5_2.html).

    My understanding is that eq. (50) describes particle horizons. Therefore, I understand that it is the particle horizon the one which increases violently during inflation and thus I am not able to understand the figures 3, 4 and 5.

    Anyway, it is mentioned in Wright and also Lineweaver that the observable universe is the distance given by the integral I have written. Since a(t) is included in this integral, the period of inflation (which influences a(t) enormously) should be taken into consideration...

    Any help will be welcome.

  7. Apr 29, 2004 #6


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    Some thoughts after re-reading again and again...:

    I think equation (50) in Watson's reference (http://nedwww.ipac.caltech.edu/level5/Watson/Watson5_2.html) is not a comoving distance, but a proper distance (the particle horizon as defined in my first post is multiplied by the scale factor in this eq.(50)).

    If one removes the scale factor multipliying the integrals in eq. (50), one gets the particle horizon as comoving distance (?) - or at least as in the definition in my first post. In this case the left side is not longer bigger than the right side, I guess (the integral would result in ~ exp[-H trec], and not ~ exp[H trec]).

    Thus, (I guess) the inflation period does not contribute significantly to the integral of the particle horizon (contrary to my claim in the first post) and therefore to the visible universe. This implies that the particle horizon, as a comoving distance, does not really grow during inflation.

    But now I am not really sure about the meaning of the proper distance here... and I am not sure about the meaning of a comoving distance for calculating the size of the observable universe as e.g. in Wright's page (I would guess a proper distance is more appropiate)...

    Last edited: Apr 29, 2004
  8. May 3, 2004 #7


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    In the meanwhile I have thought about this problem but I am not sure to have found an answer. I think I can formulate the question more precisely and I will explain my hypothesis.

    Consider a radial light ray. According to the Robertson-Walker metric it follows a path

    [tex]c dt = a(t) dr [/tex]

    Thus, the comoving distance travelled by the light ray from the big-bang to a time t0 will be

    [tex] r(t_0) = \int_{0}^{t_0} \ c dt / a(t) [/tex]

    Transforming into a proper distance

    [tex] D(t_0) = a(t_0) \int_{0}^{t_0} \ c dt / a(t) [/tex]

    This is the value of the particle horizon for a given time.

    For a matter dominated era

    [tex] a(t) = t^{2/3} [/tex]

    and, therefore, resolving the integral and calling Dm for the matter dominated part (and with the simplification considering the whole history of the universe matter dominated):

    [tex] D(t_0) = 3 c t_0 [/tex]

    On the other hand, for a period with exponential expansion (inflation)

    [tex] a(t) = e^{H t} [/tex]

    and resolving the integral calling De for the exponential expansion horizon and te the end of this period one gets

    [tex] D(t_e) = \frac{1}{H} (e^{H t_e} - 1) [/tex]

    For a reasonable small value of te, De >> Dm.

    Thus, the particle horizon (as a proper distance) during inflation grows far more than afterwards.

    Although the universe goes through a period of inflation te before of the matter dominated era until now t0, the particle horizon is calculated as Dm (Wright, Lineweaver, Watson, etc.), and not De + Dm.

    My interpretation is that Dm is the path of light AFTER inflation. If one could take a look far away towards inflation one would then be able to go additionally very far through spacetime, since the particle horizon would expand then as De. A very correct way of calculating the total particle horizon should consider also De, i.e. De + Dm, I think.

    For inflation it is important that Dm << De such that any light ray originated after inflation does not reach the border of the inflated bubble in the time of the age of the universe and thus remains within homogeneity and causal patch.

    I hope this is correct, ... it would be nice to see some comments.

    Last edited: May 4, 2004
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