How do gravitons mediate the force of gravitation in universe ?
Gravitons are hypothetical.
And "how" is not something physics can answer on a fundamental level - and that question is very fundamental.
Gravity is caused by the spacetime curvature. There are also waves of the spacetime curvature, the gravity waves. These are two forms of spacetime curvature. They are quite different.
The "gravity" (force-causing) curvature is non-wave-like, may be static and does not carry kinetic energy.
The "wave" curvature is never static, is wave-like (satisfies some wave equation) and does carry kinetic energy.
From the QM standpoint the gravity waves are some elementary particles. We call them gravitons. This is due to the particle-wave duality. Every particle is a wave, every wave is a particle. In case of gravitons this is of course a hypothesis, since we had to extrapolate QM to the GR realm and we don't know if QM still holds at that energy level.
Now, to your question. Mathematically, you can express non-wave-like curvature as a weighted sum of different gravity waves, at least to the first order. This is also true the other way - you can express a gravity wave as a sum of non-wave-like spacetime deformations, to the first order. This is a purely mathematical trick. But it QM it's a basis of some important theorems. We can deduce much about the gravitational force knowing it can be rewritten as a sum of gravitons. It doesn't mean that there are some actual gravitons flying here and there. It's just a part of QM formalism.
We don't know if the above construction is sound. Physicists invented it as an analogy to the quantum electromagnetism. This approach turned to be very useful. The force between charged particles, mediated by the electromagnetic field was expressed as sum of photons, the electromagnetic field quanta (waves). Maybe this is also the case with massive particles and the gravity field (the spacetime curvature).
For ordinary gauge theories like QED and QCD there are so-called physical gauges (e.g. Coulomb gauge) where gauge fixing = solving the Gauß constraint introduces a "potential". That means that to zeroth order no virtual particle exchange is required to explain the interaction. E.g. the el.-mag. force is described via a 1/r potential; photons are required for corrections only.
Yes, and in the original (1926) formulation of the quantum theory of radiation by Dirac, the electromagnetic field was separated into a radiation field and a static Coulomb interaction. The radiation field was then subjected to the usual quantum procedure, while the Coulomb interaction was treated as an unquantized classical interaction potential.
And this can be made exact, at least for abelian gauge theories; for non-abelian gauge theories the "potential" turns out to be a non-local, gauge-field dependent operator; anyway - it is NOT something like the exchange of perturbative virtual particles.
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