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I have a couple of naive questions from the topic of the title.

  1. We know \begin{eqnarray} \Omega-1=\frac{k}{a^2H^2}-\frac{\Lambda}{3H^2} \end{eqnarray} Now I read that from the standard big bang (SBB) model $\frac{1}{aH}$ increases with time and in recent time $\Omega\approx 1$. So, $\Omega$ has to be fine tuned to stay close to 1. I guess its because as we go back in time $\frac{1}{aH}$ decreases in SBB, and for fixed $\Lambda$ this drives $\Omega$ away from 1. I guess that's o.k as $aH$ decreases faster than $H$ alone in the second term above. But why do we want $\Omega\approx 1$ in earlier times?

My confusion deepens after reading how inflation solves this problem. In "Inflation", it is argued that $\frac{1}{aH}$ decreases with time. So I thought to myself As before going back in time resulted in decrease in Hubble length (comoving), so in the above equation the 1st term was large that is effectively with non-zero $k$. But now as going back in time increases $\frac{1}{aH}$, the first term decreases and it acts as if $k$ is nearly zero which is the criterion for flat universe. So my confusion boils down to the question on Why do we want $\Omega\approx 1$ in earlier times? Also how does Inflationary era solve the whole flatness problem? What about when inflation ends and SBB begins? Moreover, the above equation is valid for every case (inflation and SBB). So, even if inflation kills the problem, SBB should revive it yielding non unity of $\Omega$ now!

  1. Also I came to know that as long as $\dot{\phi}^2<V(\phi)$, inflation takes place and this is generally the case when potentials are flat enough (with $\phi$ of course). How to see that?

Related to that, I learned in Hybrid Inflation, since $\phi$ is driven to zero for $\psi>1$ (Why??), the potential in $\psi$ direction is flat and satisfies slow-roll conditions, so that $\psi$ is considered inflation. How can we say potential in $\psi$ direction is flat?

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the normal cosmological evolution dominated by matter or radiation - which was the case for billions of years after the Big Bang - makes $|\Omega-1|$ increase with time because of changes captured by the Friedmann equations. However, WMAP and others show that $|\Omega-1|$ is not greater than 0.01 today.

It follows that $|\Omega-1|$ had to be even much smaller a minute - and a fraction of second - after the Big Bang - something like $10^{-{\rm dozens}}$. This is called the flatness problem because there's no reason for a generic Universe to have such a small number of a quantity such as $|\Omega-1|$ which can a priori be anything.

The cosmic inflation solves the flatness problem because it reverses the evolution $|\Omega-1|$: as time goes to the future, $|\Omega-1|$ is (more precisely, was) decreasing in this case. It's because the evolution is driven by the last, temporary cosmological constant term in your equation. So because of inflation, one may start with a generic value of $|\Omega-1|$ before the inflation, and one ends up with $|\Omega-1|$ close to zero at the end, anyway.

I hope that I don't have to explain that if a quantity increases/decreases with time, it will decrease/increase if you read the time in the opposite direction. Nevertheless, it seems that 1/3 of your question is focusing on this triviality.

$\dot\phi^2 < V (\phi)$ is just a condition saying that $V(\phi)$, the cosmological constant term, is the dominant term in the Friedmann equation during inflation. If that's so, you may neglect the kinetic term $\dot \phi^2$. I am not sure whether the coefficient in the equation is one - I doubt so. You should only view the inequality as an estimate. If the kinetic term is much smaller, it's guaranteed that you can neglect it and the exponential inflationary expansion dictated by $V(\phi)$, the temporary cosmological constant, follows.

Concerning hybrid inflation, obviously, you can't determine that the potential for $\psi$ is a slow-rolling one just from the assumptions you have shared with us: it's just an assumption that the potential has the property. In particular, the potential is supposed to resemble the potentials in the theory of second order phase transitions - which occurs very generically in string theory models etc. Imagine that $\phi$ is the temperature, more precisely $T-T_c$, and $\psi$ is the magnetic field. For a positive $\phi$, the minimum is at $\psi=0$, so only $\phi$ is changing. However, once $\psi$ reaches zero, you get a phase transition and the right magnetic field is at a nonzero value, so one rolls the channel at $\psi=\pm M$. Near the $\phi=0$ point, the potential is inevitably changing slowly because it's a stationary point. See e.g.

http://nedwww.ipac.caltech.edu/level5/Liddle/Liddle5_7_2.html

The purpose of the other fields in hybrid inflation is to stop the inflation.

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This is a great answer. My compact summary of the answer to the first question: we don't want $\Omega\approx 1$ at early times; the equations force us to conclude $\Omega\approx 1$ at early times. The flatness problem is why the Universe chose to be born that way. –  Ted Bunn Mar 21 '11 at 13:54
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Its indeed a great answer. Thanks! Lots of things got clear. –  user1349 Mar 23 '11 at 4:00
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Re the $\dot\phi^2<V(\phi)$ condition - the coefficient is indeed one, since it's equivalent to requiring that $\rho+3p<0$, where $\rho$ is energy density and $p$ is pressure, and this is what is required for accelerating expansion ($\Rightarrow$ inflation). $\dot\phi^2\ll V(\phi)$ is the slow-roll approximation, which generally makes calculations nicer. –  James Aug 28 '12 at 20:54

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