* If gravity is indeed conveyed via a graviton particle, we know that it does not couple merely to mass, because gravity bends light. It must, instead, couple to mass-energy (with an E=mc^2 interaction).
It follows from the existence of Black Holes, however, that gravitons must not couple to other gravitons, just as photons don't couple to photons (this also seems to theoretically grossly disfavor massive graviton theories). But, this is odd.
Photons couple to electric charge and lack electric charge themselves. Gravitons couple to mass-energy. Yet, surely if gravity is transmitted via bosons, they must have energy which is seemingly one of the things that they couple to. The definition of energy is the capacity to apply force to move things, which gravitons, if the exist, can surely do.
FWIW, I'm not terribly clear on whether gravitational energy self-gravitates in GR itself. If it doesn't it isn't clear to me how matter-energy conservation is not violated, however. Discussions of this issues can be found here and here with seemingly with a contradictory conclusion here. More discussion here (gravity self-gravitates but does not generate gravity the way that other matter-energy does in the equations).
* Both quantum gravity transmitted by gravitons and ordinary classical GR gravity, propogate at the speed of light, not instantaneously. So, the gravitational interactions of two objects can be decomposed into two parts. The generation of a gravitational pull from a source object that has an impact at its destination (both proportionate to mass-energy apparently), and the destination object's pull in the other direction. This is sometimes called a distinction between active and passive gravity. For objects in motion the source object and the destination object gravitational impacts on each other are at a time-gap to each other.
* The coupling of gravitons to mass-energy, if they exist, is also odd because it does not always couple with the same strength to a particle of a particular type. The coupling of a photon to an electron or W boson is always identical. The coupling of a photon to an up type quark is always identical. The coupling of a photon to a down type quark is always identical. The strong force coupling of a gluon of one of the eight types of gluons to any of the six flavors of quark with a particular color charge is always identical. The coupling of a W boson or Z boson to a fermion is always a function of that kind of fermion's weak force charge.
* The Higgs boson coupling constants are such that the "sum of the square of each of the fundamental boson masses, plus the sum of the square of each of the fundamental fermion masses, equals the square of the Higgs vacuum expectation value to a precision of 0.012%."
In other words, the sum of the Yukawa's giving the proper definition of the Higgs boson self-coupling, equals one (empirically true, but not theoretically required by the Standard Model), i.e. 2λ +g2/4+(g2+g'2)/4+sum over all fermions(yf/2)= 1, where λ, g, g′ and yf being, respectively, the eﬀective and renormalized scalar (i.e. Higgs self-coupling), gauge (i.e. W and Z boson couplings) and Yukawa couplings of the twelve Standard Model fermions to the Higgs boson.
But, while the Higgs boson coupling to a fundamental particle, i.e. it's "Higgs charge", also called its Yukawa, does not come in integer or simple integer ratio units the way that electric charge, weak force charge, and strong force color charge do, the Yukawa does not.
* The electromagnetic coupling constant, the weak force coupling constant, the strong force coupling constant and the Higgs field properties, as well as the masses of the fundamental particles, all "run" with the energy level of the interaction. The mass of a quark in a low energy interaction is not the same as the mass of that same quark in a high energy interaction. But, while masses and bosonic coupling constants do run, electric charge, weak force charge, color charge do not (see e.g. here).
Still, any given kind of particle at any given energy scale, always has a particular mass. Indeed, particle mass is indifferent to (1) whether a particle is ordinary matter or antimatter (something that flips the electric charge of a particle to the opposite charge), (2) it is indifferent to its parity (which impacts its weak force charge), and (3) it is indifferent to the color of a quark (all charm quarks, for example, have the same mass at a given energy level, without regard to whether it has red, blue or green strong force color charge).
This is not true in the case of gravitons. For example, in general relativity, an electron travelling at 0.1 times the speed of light and an electron travelling at 0.5 times the speed of light give rise to different gravitational effects because they have differing amounts of kinetic energy, and the direction of the gravitational pull is not a function just of the location of the electron, as it would be in the case of Newtonian gravity who gravitons would be spin-0 bosons transmitting a scalar gravitational field, but also of the direction in which the electron is traveling. The fact that a particle's vector momentum, vector angular momentum, and vector photon flux, as well as its scalar rest-mass and other elements all contribute to gravitational pulls in general relativity is why a general relativity graviton would have to be a spin-2 boson giving rise to a tensor field, rather than the spin-0 boson of a Newtonian gravity or the spin-1 vector bosons of electromagnetism, the weak force and the strong force.
Note that the cosmological constant of GR can be conceptualized as a scalar field, however, possibly with its own spin-0 boson.
* This particularly problematic because the absolute among of energy of a matter-energy field is not a well defined universal quantity even for a particular matter-energy field. Kinetic energy is a function of velocity which is a function of the reference frame of the person describing it. So is potential energy in a variety of fields. Discussion of the arguable non-gravitation of potential energy is found here. General relativity copes with this problem by being formulated mathematically in a background independent way that essentially dependents on the differences in energy between two points in the GR field, neatly cancelling out differences in intermediate quantities like absolute energy level that don't produce physical observables.
But, it isn't obvious to me how gravitons acting in isolation can do the same, although I suppose it can base its properties on it in reference to its source, and it in reference to its destination, using itself as an intermediate reference frame.
Maybe the Beta function running of particle masses and coupling constants with energy level solves some of these issues in the Standard Model, but it is my understanding that beta functions derive from the renormalization proceedure, and not from general relativity.
Still, a graviton, at a minimum, is engaged in a far more sophisticated interaction than any other force carrying boson. The other force carrying bosons need only respond to one universal property of a particle. The graviton must measure what we would ordinarily consider to be multiple properties of a particle at once as it interacts with it, some of which must be measured in a way that is relative to the graviton's source. No other particle in the Standard Model has properties that depend upon the source of the particle in this way - the particle itself has a tiny number of individual properties that fully characterize it regardless of its source (apart from quantum entanglement).
What a GR graviton delivers is not merely a pull in a particular direction. It can deliver elaborate twisting and turning.
* One way that a Higgs boson is often conceptualized is as something that gives rise to the inertial mass of fundamental particles. But, in principle, at least, it seems as if the inertial mass of a fundamental particle due to its Higgs field interaction may differ from the gravitational mass of that same fundamental particle which derived from both the mass and the energy of the particle, seemingly violating the principle of equivalence (although perhaps equivalence merely means that the inertial mass component of a particle's mass-energy is identical to the gravitational mass component of a particle's mass-energy disregarding the energy component of particle's mass-energy).
Presumably, the running of fundamental particle energies with energy scale also impacts the gravitational mass of those particles, although it isn't obvious to me how mass-energy conservation is maintained in this context.
* Particles that don't have weak or electric charge (i.e. photons and gluons) don't have rest mass, empirically, in the Standard Model.
* The mass of a composite particle, like a proton, is not simply the sum of the fundamental particles that make it up. Those only make up about 1% of the total mass. The other 99% of the mass comes from the energy of the strong force gluon fields between the quarks in the composite particle, although the amount of the composite particle mass isn't entirely independent of the mass of the fundamental particles that go into it in a non-linear way. I've heard authoritative sources (maybe at the Of Particular Significance blog) state that even in the absence of fundamental particle mass, a proton would have mass, although it would be much less than it is in reality, which means that not all of the mass of composite particles is derivative in some way of the Higgs field interactions of the constituent fundamental particles.
* If particles in the Standard Model are truly point-like, they would be singularities in GR. But, they only need to be smeared over a sub-Planck length distance by some means to escape this fate. This is one basic issue that we would expect any quantum gravity theory to solve.
* The black hole firewall debate illustrates the deep problems involved in trying to mix classical GR and the quantum SM.
* Would hypothetical gravitons differ in energy like photons do via particle frequency (equivalent to particle wavelength), or do they have identical energy? Arguing that they do see, e.g., here and here (with a caveat since energy is not localizable in GR) and here.
The point of the observations is to reach some personal speculative conclusions and conjectures about quantum gravity.
1. The transmission of gravity via a graviton if gravity reduced to general relativity in the classical limit, then the properties of a graviton and its interactions seem much more complex and non-straightforward in their fit to a particle model than the other Standard Model interactions. This tends to disfavor particle oriented theories of quantum gravity (a la SUGRA and supersymmetry) over quantum gravity formulations that reside in an emergent space-time fabric (e.g. Loop Quantum Gravity) rather than a particle.
2. The process by which fundamental particles are endowed with inertial mass via the Higgs field is not equivalent to or identical with gravitational mass. There are gravitational mass-energies which do not have their source in the Higgs field (e.g. photons, gluon fields, kinetic energy) and the Higgs field does not generate everything that contributes to even a fundamental particle's gravitational impact. This also suggests a need for stronger experimental tests of the equivalence of inertial and gravitational mass in systems in which Higgs field generated inertial mass is not overwhelmingly predominant.
3. Is it sensible to imagine a fundamental particle (e.g. a sterile neutrino) that acquires inertial mass via the Higgs field, but lacks other Standard Model interactions? There is certainty room, even given the precision of the fundamental particle Yukawa measurements and the conjecture the the sum of all Yukawa's equals one, for this to be the case for a keV mass particle, for example. But, this doesn't fit well with a notion that the Higgs boson may in some sense be a combination of the four electroweak bosons as its mass and other properties seem to suggest (suggesting that each of the boson interactions contribute to its field and that particles that interact with none of the fields shouldn't interact with the Higgs field either, something that is otherwise true).
Note on Wiggle Room in GR Confirmation
Some of the nuances on what non-mass energy quantities are properly included in General Relativity calculations aren't very well confirmed experimentally relative to other experimentally well confirmed predictions of GR. In many circumstances in astronomy observations, energy contributions are so modest relative to mass contributions that they can be effectively disregarded, and gravitational energy contributions, for example, would be too tiny to observe for the most part.