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If the rate of expansion of the universe is increasing , won't it eventually become greater than c? Also, after inflation, did the rate of expansion slow down and then start to accelerate again?

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After inflation, the expansion of the universe did indeed slow down. During the inflationary epoch (lasting roughly $1 \times 10^{-33}$ seconds), the universe expanded by a factor of $10^{26}$. That's incredible! However, the inflationary epoch didn't last long, and that incredible expansion ended pretty soon after it started.

The universe is, as you said, still expanding. In fact (like you also said), that expansion is increasing. However, we have to be careful when we talk about the rate of this expansion. Currently, the rate of expansion between two objects depends on the distance between them, which is encoded in Hubble's law: $$v=H_0D$$ where $v$ is the recessional velocity, $H_0$ is Hubble's constant, and $D$ is the proper distance between the objects. This relation proves that objects that are farther away are moving away at a greater speed. We can extrapolate from that that, at a sufficiently far distance, two objects would be moving away from each other at the speed of light or greater! However, this is only true for objects very far from each other. But yes, eventually, any two objects sufficiently far from each other will move apart at the speed of light - and then greater.

Source (for length of inflationary epoch): http://www.universe-galaxies-stars.com/Cosmic_inflation.html

Source (for Hubble's law): http://map.gsfc.nasa.gov/universe/uni_expansion.html

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    $\begingroup$ Trivia: $v\sim c$ for galaxies at roughly a third of the radius of the observable universe, at redshift $z\sim 1.41$; for comparison, the most distant known galaxies are about $z\sim 8$, with some candidates at $z\gtrsim 10$. Also, tiny nitpick: inflation's not incredible on account of it being quite credible. :) $\endgroup$ – Stan Liou Sep 21 '14 at 21:35
  • $\begingroup$ @StanLiou Cool information. And I've been meaning to make that joke on some other questions for a while now. . . $\endgroup$ – HDE 226868 Sep 21 '14 at 21:41
  • $\begingroup$ With a Hubble constant of 68 (km/sec)/Mpc en.wikipedia.org/wiki/Hubble%27s_law anything further than 14.4 or so billion light years from us is already receding from us faster than light. It's no longer part of the observable universe. $\endgroup$ – Wayfaring Stranger Sep 22 '14 at 20:47
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    $\begingroup$ @WayfaringStranger: You're likely confusing some of the following concepts: Hubble sphere (recession velocity $c$, $\sim 4.3\,\mathrm{Gpc}$), event horizon (distance ideal light can travel from now to $t=\infty$), and particle horizon (ideal light from $t=0$ to now, which gives the radius of observable universe, $\sim 14.2\,\mathrm{Gpc}$). We can see many galaxies that that are receding superluminally from us; by volume, most of the observable universe is. $\endgroup$ – Stan Liou Sep 23 '14 at 0:29
  • $\begingroup$ @pabouk Meant to post this earlier - thanks for the correction; that was quite the error. Made the edit. $\endgroup$ – HDE 226868 Sep 23 '14 at 0:37

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