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The equation

$\Omega-1=\frac{k}{\dot{a}^2}$

together with the assumption that $\ddot{a}<0$

are the equations which demonstrates the flatness problem " because currently $\Omega$ is so close to unity and they mean that at early Universe $\Omega$ was closer to unity because $\dot{a}$ is a decreasing function" .

How come that we ignore that currently there is a cosmic acceleration where $\ddot{a}>0$?! ain't it possible that there is no fine-tuning of initial conditions and we just fall into an almost flat Universe?

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The universe has not been accelerating for long enough to make that much difference to the flatness problem.

The Friedmann acceleration equation is $$\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\left( \rho + \frac{3P}{c^2}\right)$$ where $\rho = \rho_M + \rho_{\Lambda}$, with $\rho_M$ the density of matter (and radiation), whilst $\rho_\Lambda$ is the energy density of the vacuum. For the standard cosmological constant $P = -\rho_{\Lambda} c^2$, so $$\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\left( \rho_M - 2\rho_{\Lambda}\right).$$ Thus the acceleration only becomes positive when $\rho_M < 2\rho_{\Lambda}$ or when $\Omega_M < 2\Omega_{\Lambda}$.

At present $\Omega_M \sim 0.3$ and $\Omega_{\Lambda}\sim 0.7$ so the expansion is accelerating. However, as recently as $\sim 6$ billion years ago, the universe was decelerating, would have had $\Omega \sim 1$ and thus $\dot{a}$ would have been larger before that. At those earlier times then dark energy density would have been even less important compared with matter density and so the assumption that $\ddot{a}<0$ would have been even more correct and hence the flatness problem remains.

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  • $\begingroup$ I think my confusion that you assumed that the value of $\Omega$ is equal to unity 6 billion years ago. Can we measure it back that time? $\endgroup$ – maha Dec 27 '16 at 8:09
  • $\begingroup$ @maha It cannot be far from unity given the values of $\Omega_M$ and $\Omega_{\lambda}$ we see now. If we assume a cosmological constant and the usual Friedman equations, then the values are determined at all times (with appropriate uncertainties). Measurements of distant supernovae are effectively sampling the universe 6 billion (and more) years ago. $\endgroup$ – Rob Jeffries Dec 27 '16 at 8:20
  • $\begingroup$ In theoritical basis if we have the governing equations (Friedmann's) and the initial conditions (0r boundary conditions) then we have the evolution of the unkowns. And observationally we actually have the values of $\Omega$ back then. Did I get it right? $\endgroup$ – maha Dec 27 '16 at 8:48
  • $\begingroup$ @maha When cosmological parameters are measured, then usually you are measuring things in the past, so you are inherently measuring the evolution of the universe, not the the instantaneous values of the parameters now. Obviously there are exceptions. $H_0$ is the Hubble parameter now. $\endgroup$ – Rob Jeffries Dec 27 '16 at 8:54
  • $\begingroup$ I'm confused about your last point because type Ia supernovae is a clue for our expansion "now". $\endgroup$ – maha Dec 27 '16 at 9:42

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