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Is the inflation speed of the universe accelerating or is it a constant speed of expansion proportional to distance between objects.

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As the universe stands today its not said to be "inflating" but it is "expanding". I think one way to make the distinction is to say that during inflation the radius of the causal horizon ($\frac{1}{aH}$ in FRW universe) falls (linearly) whereas during expansion it increases (being equal to half the conformal time for EdS universe).

I would love to know if some other answers can make it more precise.

Also about how much and for how long did inflation happen - I think Wikipedia gives a number in the first paragraph of the entry on cosmological inflation.

I would love to know if there is any theoretical derivation of these mind-boggling numbers.

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See At what speed does our universe expand? and Speed of Universe Expansion for related questions, but I take your question to be specifically asking if the rate of expansion is increasing. The answer is that the expansion rate is increasing, and this was measured experimentally in 1998 by Perlmutter and Riese's groups.

The reason for the increase is another matter. We tend to ascribe it to dark energy, but no-one knows for sure if this is a good description or what the physical origin is.

Response to comment:

We tend to work with a deceleration parameter $q$ given by:

$$ q = -\frac{\ddot{a}}{aH^2} $$

where $H$ is Hubble's constant. It's called the deceleration parameter because it was first used long before the acceleration of the universe was discovered - for acceleration $q$ has a negative value. Anyhow, to calculate the acceleration change over the radius of the Earth we use:

$$ \ddot{a} = -qH^2r_E $$

Some quick Googling suggests the current value of $q$ is about -0.63, the radius of the earth is 6,371,000m and in SI units $H$ is about 2.3 $\times$ 10$^{-18}$/sec so the acceleration works out to be:

$$ \ddot{a}_E \approx 2 \times 10^{-29} ms^{-2}$$

I can see why you asked as it would be interesting if the acceleration due to dark energy was comparable with the acceleration due to gravity. However there is around 29 orders of magnitude difference! The acceleration is only detectable at cosmological distances.

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Is there anyone out there that can calculate this apparent acceleration speed given by the Perlmutter experiment, based on the diameter of our earth as a distance? Will it be close to the acceleration of gravity per chance? – George Jones Mar 31 '13 at 2:08
    
@GeorgeJones: I've updated my answer to respond to your comment. – John Rennie Mar 31 '13 at 8:10
    
thank you for the response. – George Jones Mar 31 '13 at 16:48
  • When we talk about universe inflation we should have in mind that the four cosmic forces did not appear yet, because at inflation no elements were formed yet.
  • scientists say that at 0.03 second the inflation reached four light-years, also said that if it had continued at the same rate, it would have been disappeared during the second have of the first second.
  • before the Big Bang the inflation speed was zero. And time is zero.
  • now we can say: Light travels during four year = 300,000 km/s x 60 s x 60 min. x 24 hours x 365 days x 4 years = 37.843 x 10ᴧ12 km, this distance was within 0.03 of first second of Big Bang, means 37.843 x 10ᴧ12 km / 0.03 seconds = 12.6 x 10ᴧ14 km / Sec. Thus, inflation speed at the peak (before the effect of the four cosmic forces) in regard to speed of light is = 12.6 x 10ᴧ14 / (3 x 10ᴧ5) = 4.2 x 10ᴧ9 times the speed of light. -s = v 1 * t + 1/2 a * tᴧ2: s: distance at certain time = 37.843 x 10ᴧ12 km. v 1: speed at zero time = 0. a: acceleration. t: time = 0.03 second. 37.843 x 10ᴧ12 km = 0 + 1/2 a * 0.03ᴧ2 a = (2 * 37.843 x 10ᴧ12) km / 0.03ᴧ2 = (75.686 x 10ᴧ12) / (9 x 10ᴧ-4) = 8.41 x 10ᴧ16 km/sᴧ2.
  • the acceleration at universe inflation stage is: 8.41 x 10ᴧ16 km/sᴧ2
  • When the elements were formed starting with Hydrogen, the inflation speed came down due appearance of the four cosmic forces that started action, and the universe expanding started with much lower speed than the speed of light.
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