I'm having a problem understanding the resolution of the flatness problem in the standard Big Bang model (k = 0 and Omega = unity).(adsbygoogle = window.adsbygoogle || []).push({});

If k=0 at the present time, then this implies Omega ~ unity (actual mass/energy density ~ critical mass/energy density).

For k = 0 and H0 (Hubble parameter for present time) = 71km/s/Mpc then rho(critical) = 3.H^2/8.pi.G = 9.4665 x 10^-27 kg/m3.

BUT ~73% of this critical mass/energy density (at the present time) is supposed to come from the vacuum energy (Dark Energy) which is hypothesised to be scale-invariant (the vacuum energy density scales as a^0, in other words it does not change as the universe expands). This implies a vacuum energy-density of 6.91 x 10^-27 kg/m3.

Most of the remaining 27% of mass/energy density (at the present time) is made up of matter (visible and cold dark matter), the density of which scales as a^-3, ie the matter density scales as the inverse cube of the size of the universe). This implies a matter-density of 2.56 x 10^-27 kg/m3 at the present time.

The contribution from radiation energy-density at the present time is less than 0.01% of the total.

The above implies that when the universe was one tenth of its present size (which, because a scales as t^2/3 during the matter-dominated era, was when the universe was about 3.16% of its present age, or about 400 million years old) then the mass-density was 1,000 times greater than it is now, which implies a matter mass-density ~2.56 x 10^-24 kg/m3. The vacuum energy density contribution would have been the same as now, at 6.91 x 10^-27 kg/m3. The contribution from radiation energy-density at that time would have still been very small (~0.3% of the total). Thus matter contributed ~99.4% of the total energy density, and the total energy density would have been ~2.58 x 10^-24 kg/m3.

Assuming flatness (k=0), what would have been the critical mass/energy density when the universe was one tenth of it's present size? The same equation applies, rho(critical) = 3.H^2/8.pi.G. But at that time H would have been much higher than it is now. In fact, during the matter-dominated era H scales as t^-1 (this follows from the definition of H as a'/a, where a' is the expansion velocity; during the matter-dominated era a scales as t^2/3 and a' scales as t^-1/3). Therefore H would have been ~H0/0.0316 = 2,247 km/s/Mpc. This gives a value for the critical energy density at that time of ~9.48 x 10^-24 kg/m3.

But the actual mass/energy density, from above, was only ~2.58 x 10^-24 kg/m3, which is only ~27% of the critical mass/energy density at that time, hence Omega = 0.27 when the universe was only 400 million years old.

Where am I going wrong?

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# Flatness problem?

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