Particle creation, energy density and the Compton wavelength

[jcap]

The Compton wavelength of a particle is given by
$$\lambda=\frac{h}{mc}.$$
One can construct an expression for the energy density $\rho$ of a particle of mass $m$ given by
$$\rho = \frac{mc^2}{\lambda^3}=\frac{m^4 c^5}{h^3}.$$
What is the physical significance of the mass scale $m$ in the above expression?

Does it mean that particles of mass $m$ will spontaneously appear when the energy density $\rho$ reaches the relevant level as the Universe cools?

Is the expression only correct for fermions as it assumes only one particle (or more correctly one particle/antiparticle pair) per $\lambda^3$ volume?

Once the energy density $\rho$ cools to a level such that $m$ is less than the mass of any particle would one then get massless particles such as photons spontaneously produced with an energy $h\nu=mc^2$ ?

 

[mfb]

Does it mean that particles of mass $m$ will spontaneously appear when the energy density $\rho$ reaches the relevant level as the Universe cools?

Things cannot spontaneously appear. There can be pair creation, and in general it gets more likely with higher pressure. It doesn't necessarily mean it has to happen. If we plug in 1 eV as an upper bound on neutrino masses we get 0.08 Pa, that is a good vacuum, with no process that would produce any relevant number of neutrinos if the pressure comes from air.