I have proposed the above question to stimulate discussion. However, I start with an opinion which is that gravitons do not exist. In a paper to be published in April, (preprint available at:the link below) I show that there is a previously unknown simple relationship between the electromagnetic force and the gravitational force which would not exist if the forces were carried by gravitons and virtual photons. A brief description will be given here, but see the paper for a more complete description.

The natural unit of distance for a fundamental particle is its reduced Compton wavelength designated: λc = ħ/mc. Suppose that we compare the electrostatic force (FE) and the gravitational force (FG) between two of the same particles when they are separated by their reduced Compton wavelength λc. These forces will be expressed in natural units where c = G = ħ = 1. This results in force being a dimensionless number equal to the force expressed in SI units divided by Planck force. Also for this comparison of forces we will assume that both particles have Planck charge which is chosen because it has the fundamental coupling constant equal to 1 rather than elementary charge e which has a coupling constant equal to the fine structure constant α. At separation distance r = λc, the relationship between these forces is a simple square: FG = FE2. At arbitrary separation distance the relationship is FGN2 = (FEN2)2 where the separation distance r is expressed as N times the reduced Compton wavelength (r = Nλc). Therefore even at arbitrary distance the relationship between these forces can be expressed as a square. Also, even if the electrostatic force is a negative number, the square is always a positive number – gravity has only one polarity. This relationship fits perfectly with a wave-based explanation of forces and is incompatible with gravitons.

http://onlyspacetime.com/QM-Foundation.pdf

This is more of an advertisement than a question so I'm not sure it's really appropriate.

Stephen:     I have always assumed that a question on Research Gate can be proposed to stimulate discussion and information exchange. The question "Do gravitons exist?" introduces a debatable topic which has many supporters on each side. The answer that I proposed introduced simple equations which show that the gravitational force can be expressed as the square of the electrostatic force when both forces are expressed in the natural units of fundamental particles. This is a new insight that is independent of the theory used to derive this relationship. One of the conclusions that can be drawn from this new insight is that gravitons cannot explain this relationship. It was my opinion that the short post required the inclusion of a reference to provide additional details.  I hope that this reference is appropriate.   

I'm new to this Research Gate thing, so maybe I'm doing it wrong, but it looks to me like you asked a question to promote your preprint. To at least provide an answer to your question then, it would take a radical change to dispose of gravitons.

Let me ask a question based on your post above then: If you're describing a wave-based explanation of forces, are you arguing other gauge bosons don't exist either?

Stephen,     Apparently the information that I have posted previously was not sufficient for you to analyze my contention that the gravitational and electrostatic forces are related. This is understandable and it supports my point that the referenced paper is required if I am going to prove that gravitons do not exist to a skeptical reader.

Equations 15 to 23 in the paper relate the electrostatic and gravitational forces, but for illustration I will use the simplest: FG = FE2 (equation 19). As previously stated, this equation is expressing forces in dimensionless Planck units (c, G and ħ equal 1) and assumes the separation distance is equal to the particle’s reduced Compton wavelength. For a numerical example, suppose that there are two fundamental particles, each with the mass of an electron but with Planck charge (1.88×10-18 coulomb). If these two particles are separated by their reduced Compton wavelength (3.86×10-13 m), then the electrostatic force between them in dimensionless Planck units would be 1.75×10-45 and the gravitational force would be 3.07×10-90, (note the square). This square relationship occurs at arbitrary separation as previously explained, if we express separation distance as N, the number of reduced Compton wavelengths.

The paper derives these equations from a wave-based model of fundamental particles. However, since these equations express relationships between forces, they are independent of the model originally used to derive them. The obvious points are that the gravitational and electrostatic forces fundamentally scale as a function of the reduced Compton wavelength and are related through the square of a wave amplitude term. These relationships are undeniable and they are incompatible with gravitons and virtual photons serving as force carriers. I do not want to expand this discussion to gluons, W and Z bosons in this restricted communication. Virtual particle pair fluctuations in the vacuum are a key part of this model but gravitons and virtual photons as force carriers are rejected.

All of the gravity fields I am aware of arise by actions of discrete quantized sources. In that regard, the gravity fields must be also quantized, but on such a small scale that continuum mechanics describe most actions very well. The study field of quantum gravity has tried to decide whether every particle supports a gravity field, or a collection of particles is needed. Results are inconclusive.