Flatness Problem

Today, the Universe has Omega = 0.1 - 0.3 which is roughly 1. Huh? Why do I say that 0.1 ~ 1 and why is this a problem? To understand, recall that

-k = 1/2 [dR(t)/dt]**2 - [(4 pi/3) G rho(t) - CC/3] R(t)**2

Now, for the sake of argument, suppose that the Cosmological constant (CC) is 0. We then have

-k = 1/2 [dR(t)/dt]**2 - (4 pi/3) G rho(t) R(t)**2

Let us define a general Hubble's law, dR(t)/dt = H(t) R(t). This is a general expression because we do not specify that we measure the Hubble constant today. Since the Universe expanded faster in the past, the Hubble constant is not really a constant and was larger in the past. Also, since the density changes as the Universe expands, Omega, the ratio of the density to the critical density, also changes as the Universe evolves. Let us investigate this situation.

Divide the above relation through by R(t)**2 so that

-k/R(t)**2 = 1/2 H(t)**2 - (4 pi/3) G rho(t)

Now, divide through by (4 pi/3) G

-k 3/{4 pi G R(t)**2} = rho(crit) - rho(t)

===> rho(t)/rho(crit) = 1 + 2k/{H(t)R(t)}**2 ===> Omega(t) - 1 = 2k/{H(t)R(t)}**2

Hmmmm. What are to make of this monstrosity? Well, there are several things we can say.

• Since {H(t)R(t)}**2 is always positive (the square of any real number has to be be greater than or equal to 0)
  ---> Omega(t) - 1 is always less than 0 if k less than 0

  ---> Omega(t) - 1 is always equal to 0 if k = 0

  ---> Omega(t) - 1 is always greater than 0 if k greater than 0

  That is, in an open universe, Omega is always less than 1 and in a closed universe, Omega is always greater than 1!! Universes do not make transitions from to open to closed and vice versa.

• Because the Universe is slowing down, H(t)R(t) = dR(t)/dt, the expansion rate, was greater in the past. This means that the value of Omega - 1 gets smaller as the Universe expands.
  • So, how does Omega(t) - 1 evolve? It turns Omega(t) changes a great deal. In order for Omega(t) to be around 0.1 - 0.3 today, Omega(t) must be very close to 1 in the early Universe.

    Omega(t) - 1 = k/[H(t)R(t)]**2 = k/[dR(t)/dt]**2

    For universes dominated by radiation, R(t) grows roughly as sqrt(t) at early times ---> Omega(t) - 1 ~ k/time and the difference shrinks as the Universe evolves!

    • Omega(t) must be ~ 1 at the Planck time to 60 digits or so