# Size of universe after inflation?

Wikipedia states the period of inflation was from $10^{-36}$sec to around $10^{-33}$sec or $10^{-32}$sec after Big Bang, but it doesn't say what the size of the universe was when inflation ended. Just saw a Brian Greene show on the Multiverse and I thought I heard him say size was galactic scales when inflation ended. However I've also read size was about a basketball.

Are there multiple theories with different resulting sizes? Does 'size' even mean anything in this context?

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Unless I'm missing something there doesn't seem to be a consensus. However the question still seems sensible to me- at the time of the big bang everything was really tiny and it started expanding. Why can't you know the volume before inflation and after? –  Art Hays Jul 27 '12 at 17:49

With the proper definition of the "size" of the universe, this question does make sense. The standard model of cosmology would say that the universe is infinite which therefore does not have a "size". However, if we take into account that the big bang occurred $13.7 \pm 0.17$ billion years ago we can define a meaningful size for the observable universe. You might, for example, define the size of the observable universe as the distance a photon could have traveled since the big bang.

Consider, for example, a cosmic microwave background (CMB) photon that was emitted as visible light about 379,000 years after the big bang and is just now hitting our microwave detectors (the redshift is z=1089): that photon has been traveling for 13.7 billion years so it has traveled a distance of 13.7 billion light years. So you might imagine that the current radius of the observable universe is 13.7 billion light years. However, during this time the universe has been expanding, so the current position of the matter that emitted that photon will now be 46.5 billion light years away. (By now, the little $10^{-5}$ bumps on the CMB will have condensed into galaxies and stars at that distance.) This gives a diameter of the current observable universe of 93 billion light years. Note that as time passes, the size of the observable universe will increase. In fact it will increase by significantly more than two (to convert radius to diameter) light years per year because of the continued (accelerating) expansion of the universe. Also note that we will not be able to use photons (light) to explore the universe earlier than 379,000 years after the big bang since the universe was opaque to photons at that time. However, in the future we could conceivably use neutrinos or gravitational wave telescopes to explore the earlier universe.

So given a size of the current observable universe, we can ask how big was that volume at any particular time in the past. According to this paper at the end of inflation the universe's scale factor was about $10^{-30}$ smaller than it is today, so that would give a diameter for the currently observable universe at the end of inflation of 0.88 millimeters which is approximately the size of a grain of sand (See calculation at WolframAlpha).

It is believed that inflation needed to expand the universe by at least a factor of 60 e-foldings (which is a factor of $e^{60}$). So using WolframAlpha again we find that the diameter of the universe before inflation would have been $7.7 \times 10^{-30}$ meters which is only 48,000 Planck lengths.

Perhaps Brian Greene was talking about the size of the observable universe at the time when the CMB photons started traveling towards us. That happened 379,000 years after the big bang at a redshift of 1098 which means the universe was about 84.6 million light years in diameter which, per WolframAlpha, is about half the diameter of the local super cluster of galaxies or about 840 times the diameter of our galaxy.

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Nice answer. Also, according to the APOD scales of the universe (apod.nasa.gov/apod/ap120312.html), a human ovum is more like 0.12 millimeters. Not that it matters... I just like the applet. :) –  AdamRedwine Jul 26 '12 at 19:20
@AdamRedwine Thanks. Wikipedia agrees with APOD so I changed the comparison to a grain of sand. –  FrankH Jul 26 '12 at 22:32
This answer is using a bad notion of the radius of the observable universe. You should measure the radius along a past light cone, without extrapolation to "now", extrapolation is not the right way to describe the physics. The universe was 380,000 light-years across at the time of photon decoupling, measured along the back light-cone, and this is the physical size, and it's the size of a galaxy. The "now" extrapolations are unphysical and arbitrary. –  Ron Maimon Jul 27 '12 at 4:50
@RonMaimon - Sorry, but I disagree. We are detecting CMB photons that were emitted by real atoms and while the photons were on the way here for the last 13.7Blyr, we can confidently say that those atoms have formed galaxies and stars that are now about 46Blyrs away from us. Taking the scale factor a(t) back in time to when the universe was 379,000 years old the universe was 1/1098 of it's current size so those same atoms, at that time, were spread out over a volume with a radius of 42Mlyrs. (...cont'd...) –  FrankH Jul 27 '12 at 18:47
@Monkieboy - Where did you get 400000 years of inflation? Inflation ended at about $10^{-32}$ seconds after the big bang - that is when the universe was only 0.88mm. The 400000 years age is when the universe had expanded and cooled enough to become transparent - that is when the CMB photons started traveling towards us 13.7 billion years ago. At that time the universe was about 42 million light years in diameter. –  FrankH Oct 22 '12 at 15:37

In the simplest model of the universe, the FLRW metric, the universe is infinite and has always been infinite right back to the Big Bang. Inflation doesn't change this assumption.

So it makes sense to ask, for example, how big a Planck volume became during inflation, but it doesn't make sense to ask how big the whole universe is. (Depending on what you take as the inflation scale factor a Planck volume ended up about $10^{-27}m^3$ and this is a lot smaller than a basketball.)

Having said this, Don Page has suggested a lower bound for the size of the whole universe at the end of inflation, and his answer is $10^{10^{10^{122}}}$ cubic megaparsecs. However I think you should regard this as extremely speculative.

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How would you know it's infinite? The FLRW metric is only a predictive metric for this patch, and any extension past the horizon is speculation about unobservable things. –  Ron Maimon Jul 27 '12 at 4:51