Quantum fluctuations in early universe

[spaghetti3451]

Why do quantum fluctuations of fields arise at high energies and temperatures?

What is the mathematical formulation of these quantum fluctuations?

Why are the sizes of these quantum fluctuations approximately the Planck size?

 

[PeterDonis]

Why do quantum fluctuations of fields arise at high energies and temperatures?

Quantum fluctuations of fields are present on all energy/temperature scales. They don't happen only at high energies/temperatures.

Why are the sizes of these quantum fluctuations approximately the Planck size?

Quantum fluctuations are present on all length scales, not just the Planck scale.

 

[Chalnoth]

Why do quantum fluctuations of fields arise at high energies and temperatures?

It's actually the reverse.

Quantum fluctuations always occur, no matter the temperature. But they are only really significant at extremely low temperatures. The quantum fluctuations that laid the seeds of structure in our universe formed during an era where the temperature was pretty much as close to absolute zero as you could get.

What is the mathematical formulation of these quantum fluctuations?

The particular fluctuations you seem to be talking about are zero-point fluctuations of the inflaton field. If you want a really in-depth look, here's one source:
http://www.damtp.cam.ac.uk/user/db275/TEACHING/INFLATION/Lectures.pdf

Why are the sizes of these quantum fluctuations approximately the Planck size?

I think that's more of an assumption than anything. How big they were depends more upon the properties of inflation: the fluctuations would have been close to the size of the horizon at the time of inflation. But we don't know what that horizon size was yet because we don't know for sure what the energy scale of inflation was.

 

[bapowell]

Why are the sizes of these quantum fluctuations approximately the Planck size?

If by "size" you mean wavelength, then there is no assumption that they must be Planck size, though that is generally how they are treated (i.e. you can have a quantum fluctuation of any wavelength). The thing is that larger wavelength fluctuations spend less time within the horizon during inflation and so are amplified much less than Planck sized fluctuations. How to evolve the fluctuation from sub-Planckian to super-Planckian scales is known as the "trans-Planckian problem", since, lacking a UV-complete theory of gravity, we don't actually know how sub-Planckian fluctuations evolve.