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Once again, an attempt to connect popular and real science.

I have read that inflation implies our universe should look flat, or really close to flat, because our observable universe is only a really small part of a much larger "inflationary bubble". So, we are like people who can see only a very small patch of the Earth and think its flat.

See, for example:

In that case, do we have any theoretical or empirical estimates of how large our inflationary bubble is compared to the observable universe ?

I read an article by Andrei Linde in Scientific American saying that the size of the bubble could be $10^{10^{12}}$ metres (1 followed by a trillion zeros). Is this generally accepted ?

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The answer depends on how long inflation lasted, how rapidly it expanded, and how big the bubble was to start with. These parameters are obviously model dependent and there are many models none of which is entirely convincing.

For arguments sake let's assume that inflation ends at the GUT scale with a doubling in size every Planck unit of time and with an initial diameter of the planck length. The GUT scale in time is 10000 planck units so the overall expansion factor is $2^{10000} \sim 10^{3000}$ . With this factor the initial size of $10^{-43}$ meters is almost irrelevant and we can say the final bubble size is around $10^{3000}$ meters.

Obviously you could make different assumptions and get very different answers. I don't know what assumptions Linde was using but I guess he was taking the fastest inflation he thought was possible to get an upper limit.

It is at least as interesting to consider the lower limit that is consistent with observation. This depends on how flat we observe the universe to be but it has to be bigger than the observed universe so not much smaller than $10^{25}$ meters

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Philip - regarding the lower limit you mentioned: Suppose the inflationary bubble was spherical (or rather, hyperspherical) and only a little larger than the observable universe. Then wouldn't it seem curved, rather than flat to us ? In my analogy, the observable universe would then be like a patch which covers a very large part of the Earth, so we can "see the curvature" –  user1265 Jan 19 '11 at 11:12
    
@Cosmonut, yes you would expect some curvature if it were just a little larger, but if we are being conservative to set a lower limit we might allow for quite a sudden change. Don't forget that there are some "holes" in the background radiation in some directions so we shouldn't be over-confident about how flat it is near the visible horizon. In any case my figure for the lower limit can probably be improved. It was only intended to be very rough. –  Philip Gibbs Jan 19 '11 at 13:30
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