Cosmic gravitational background and its temperature [duplicate]

There is a cosmic microwave background, according to the Big Bang theory. There is also a cosmic neutrino background, at 1.945 K, yet to be discovered, according to the Big Bang theory. My question is: what is the cosmic graviton/gravitational background temperature? If any, how could it be related to the neutrino or photon background temperatures?

Remark: Sometimes, some people distinguish between the stochastic gravitational wave background and the so-called primordial gravitational wave events. The answer to this question should also clarify what are these two GW and how could they be related to the hypothetical cosmic gravitational wave background.

• Since we do not have a theory of quantum gravity, or indeed confirmation that gravitons exist, this seems an underspecified question - what model in which gravitons exist are you asking about here? – ACuriousMind Sep 11 '17 at 11:08
• If gravitational waves exist, and they do, by duality wave-particle gravitons do exist. However, I am aware that maybe we can not answer this until quantum gravity is formulated. But, it is just a question I could not resist to ask. – riemannium Sep 11 '17 at 11:29
• By what "duality"? The "wave-particle duality" is a handwavy concept, not a rigorous thing you can apply to all types of wave. – ACuriousMind Sep 11 '17 at 11:37
• – Kyle Kanos Sep 11 '17 at 11:53
• Click on @ACuriousMind handle to see his bio before you try to teach him physics! – user154997 Sep 11 '17 at 13:44

It's difficult to say. As @ACuriousMind mentioned in comments, it will be model dependent. Why? Well, look into what fixes the temperature of the (CMB), made of light. The short version is that the temperature of matter and light were held in lock-step as long as matter consisted of an ionized plasma of mostly hydrogen. Once the electrons and protons combined into neutral hydrogen, an event called recombination, the light and matter were free to evolve separately, with the matter cooling much faster than the light. As a side note, the reason non-relativistic matter cools faster in an expanding universe is because the momentum of the particles goes like $1/a$ (decreases inversely with the scale factor of the universe), both classically and quantum mechanically. Since the kinetic energy of non-relativistic particles is proportional to $p^2$, the temperature of non-relativistic particles will drop like $1/a^2$, whilst relativistic particles drop like $1/a$.
The temperature at which recombination happens depends on the density of particles in the plasma, and the math (in the Wikipedia article) suggests it happened at a temperature of around $4000\operatorname{K}$.
You can tell a similar story about the cosmic neutrino background (C$\nu$B). Although, Wikipedia's "It is estimated that today, the C$\nu$B has a temperature of roughly $1.95 \operatorname{K}$," should be taken with a grain of salt because we know that neutrinos have a non-zero mass less than $0.120 \operatorname{eV}/\operatorname{c}^2$, which would mean they go non-relativistic somewhere below a temperature of $1400\operatorname{K}$. Point being, unless their mass is less than $170\operatorname{\mu eV}$, the C$\nu$B is significantly colder than $2\operatorname{K}$.