# Normal Modes on the surface of a sphere- Big Bang

I am modeling the early Universe, just before the Big Bang.

The problem maps to Normal modes on the surface of a Sphere where the sphere is expanding and the sound velocity abruptly decays at a given density.

At the same time the density decays, energy is released into the volume accordingly with the local density.

When the sound velocity is small in comparison with the radial expansion, and the energy stored per volume is totally released, the density distribution are imaged at the moment of transparency.

This problem could be solved on a sphere or around a circle.

I would start with a circle. That said, I would welcome any help on the right direction since I haven’t done fluid dynamics in the past.

Hello,

welcome on the forum. It sounds like an interesting problem.

It is a bit difficult to give you guidance in particular. But I guess you need a compressible model. Would your model also be compatible with the perfect gas law or would it be something else?

Compressibility of Neutron Stars is not required since the relevant process takes place when the Neutrons start to decay (when the density reaches 305 MeV/fm3
That corresponds to the Universe being a hyperspherical hypersurface of 800 light-seconds. Between that time and the time when density decayed to 50 MeV/fm
3, the speed of sound decayed by a factor of 3. That decay of the speed of sound freezes dynamics.

So, the modulation in density is likely to be guided mostly by the plasma dynamics and the triggering of hyperspherical normal modes. One could try to develop an equation for the transport but I think it might be easier to just seed randomly energy around and run the trajectory for the several seconds.

The mass distribution of the Universe should had been settled in few minutes because of the changes in the speed of sound.

My initial idea is just to use perfect gases to calculate the temperature at the end of the energy released. I have another formula for the further cooling (associated with the topology and paradigm for motion).

Perfect gas is a good approximation at this point.

OK. In summary you have an initial distribution of temperature, density, and momentum and will simulate from there. Correct? If that’s the case and you have large density/temperature variations you a “compressible” model.

What kind of boundary conditions will you use?