Particle horizon and inflation

In summary: Regards.In summary, calculating the size of the observable universe involves finding the current distance to the particle horizon, which can be expressed as an integral with the scale factor a(t). In order to calculate this integral, one must first determine an expression for a(t). In the case of inflation, the dependence of a(t) is different, but this only has a minor effect on the overall calculation of the observable universe. Therefore, the particle horizon does not significantly contribute to the visible universe during inflation.
  • #1
hellfire
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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?

Regards.
 
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  • #2
this is a good puzzle
I don't see how to resolve this. I am sure somethings wrong
but can't say what. I hope somebody else shows up who can
say.

Searching for help, I found a recent set of lectures on cosmology topics.
http://www.mpia-hd.mpg.de/homes/rix/Lecture.html

Lecture 4 is on inflation
http://www.mpia-hd.mpg.de/homes/rix/Inf_Lect.pdf

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
expands

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

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
 
  • #3
here's another link referring to "shrinkage" during inflation

http://nedwww.ipac.caltech.edu/level5/Watson/Watson5_2.html

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:
http://nedwww.ipac.caltech.edu/level5/Watson/Watson_contents.html

Here is a Berkeley copy of a Harvard page that links to this Watson page and has other online references about inflation:
http://astron.berkeley.edu/~jcohn/chaut/inflation_refs.html

for completeness here's a link to Ned Wright
http://www.astro.ucla.edu/~wright/cosmo_04.htm
 
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  • #4
new article by Alan Guth called simply
"Inflation"

seems to be pedagogical, partly introductory
partly survey of different scenarios
might be useful
http://arxiv.org./astro-ph/0404546 [Broken]
 
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  • #5
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.

Regards.
 
  • #6
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)...

Regards.
 
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  • #7
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 traveled 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.

Regards.
 
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What is the particle horizon?

The particle horizon is a concept in cosmology that refers to the maximum distance from which light or other particles can reach an observer in the present age of the universe. It is essentially the observable universe, as anything beyond the particle horizon is too far away for its light to have reached us yet.

What is inflation in cosmology?

Inflation is a theoretical period of rapid expansion that is believed to have occurred in the early universe. It is thought to have happened in the first fractions of a second after the Big Bang and is used to explain certain observations, such as the uniformity of the cosmic microwave background radiation.

How does inflation relate to the particle horizon?

Inflation has a direct impact on the particle horizon. The rapid expansion during inflation caused the particle horizon to expand as well, allowing for the observed uniformity of the cosmic microwave background radiation. Without inflation, the particle horizon would be much smaller, and we would not be able to observe as much of the universe as we can today.

What is the evidence for inflation?

There is strong evidence for inflation from various observations, including the uniformity of the cosmic microwave background radiation, the large-scale structure of the universe, and the flatness of the universe. Additionally, inflation is supported by mathematical models and simulations that accurately predict these observations.

Is inflation a proven theory?

Inflation is a widely accepted theory in cosmology, but it is not yet considered a proven theory. While there is strong evidence for inflation, it is still an active area of research, and scientists are continually studying and testing its predictions. Further observations and experiments are needed to confirm inflation as a proven theory.

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