Thanks for your reply.

(Even in GR, light does not in general move on null geodesics. That is only true in the geometric optics limit, it's a question of how the wavelength compares to the curvature scale.)

No debating that, but at least cosmologically light from distant objects moves on what may well be approximated as null geodesics. In a Palatini [tex]f(R_{ab}R^{ab})[/tex] theory even this would not be what was observed (although one could always choose the theory so it was approximately the case). That's all we are saying there.

The issue of internal structure is distinct from the quantum mechanical nature of particles. In QM, neither photons nor dark matter particles are strictly localized clumps: they have a wavefunction with some spread, so there is strictly speaking no "empty space".

It's true that the classical description of particles separated by vacuum is not strictly accurate, and is an idealization of the more realistic situation where one would quasi-localized clumps of energy separated by large regions of almost empty space (ignoring issues with things popping in and out of the quantum vacuum). This said we think that for many situations of interest, not least cosmology after the radiation the classical particle description is pretty good and certainly much closer to the fluid description that is standardly used.

Just consider what would have to happen for this approximation to be wrong. Cosmologically < proton per every cubic metre. So we know that the proton is not a localized clump of matter but say its energy density has gaussian distribution with width ~ 1/proton mass, and so some exponentially minute faction of the energy density in such a proton is spread over distances > 1m. Now for the relative motions of two protons (seperated by a few metres say) in the Palatini theories to actually be different from GR (in a significant way), one would have to require that the motion of proton one a few metres away from proton two could be affected in a significant way by the Palatini force caused by tiny gradients in that tiny fraction of the energy density of proton two that is spread over that distance, and so the cosmology would look different from GR. Whilst this may happen in particularly pathological theories, I think it was reasonable of us to assume that this is not the case in any theory vaguely connected with reality, and approximate matter as being in classical lumps.

At any rate, I would say that the homogeneous fluid (or even an almost inhomogeneous fluid) approximation for cosmology matter is certainly much further away from reality that the classical lump approximation we use. These lumps of matter are not assumed to have zero extent, but are assumed to have an extent that is to all intents and purposes essentially finite and much smaller than the inter-particle separation.

This is not an analogous situation. In this case, you are comparing the evolution of a perfectly smooth medium and a medium which is clumpy. .... (And this is not sensitive to the microscopic details.)

In making the analogy, I was only trying to say this: In GR the evolution of a system with one scale [tex]L_{1}[/tex] can be sensitive to the distribution of matter on scales [tex]L_{2} \ll L_{1}[/tex] if this distribution is such that the non-linearity in the GR field equations is important on scales [tex]\sim L_{2}[/tex]. If this is the case then solving the Einstein equations with an averaged energy momentum tensor would give a different answer to solving the metric w.r.t. the microscopic (that is over scales < [tex]L_2[/tex]) energy momentum tensor and then averaging the metric.

Precisely the same thing is happening with these Palatini theories. Here the way in which Palatini gravity deviates from GR is governed by a algebraic field equation that depends on the local density of matter and not the spatial extent of it. This equation links the R in the f(R) to the trace of the energy momentum tensor. To the extent that this field equation is linear GR and Palatini are the same, so all the deviations are encoded in the non-linear terms. As a result deviations of Palatini from GR are at all scales determined by a non-linear algebraic equation which depends on the local energy density. No matter how big your clumps of matter are, if you average the local energy density, you will therefore get the wrong answer. It's because of this non-linearity one all scales that the distribution of matter on microscopic scales matter.

In GR this doesn't occur as if you go to small enough scales everything linearizes nicely, but on the scales where the non-linearities become important then averaging breaks down and you have to be more careful.

It was in that sense that I made the analogy.

The key point though is that in all Palatini theories the effective extra force you introduce (which is essentially the modification to gravity) has zero range. So if you have two particles whose energy densities (to a good approximation) do not overlap, the would not feel any extra force between them. As such their relative motions are unaffected by this extra force. As it well known Palatini theories are such GR with a modified source. The modified source means that you may (and do) alter electron energy levels in atoms (since effective electron mass now depends on the local electron density) or modify nuclear binding energies or other such things. But independent of the internal distribution of energy and momentum in a microscopic particle (or clump of energy momentum), in GR all such clumps move in essentially the same way. If no extra long range forces are present then the particle such move as classical particles under gravity, and so cosmologically they look to a good approximation like dust or as a better approximation collisionless matter.

To change this state of affairs one would have to introduce a long range force, or at the very least a force with non-zero range, and in Palatini theories you don't have one. In metric [tex]f(R)[/tex] theories, there is such a force, which is why they are so tightly constrained by fifth force experiments.

It should be pointed out however that one would still get a cosmological evolution that looks like something one would predict from Palatini with a perfectly homogeneous matter distribution, but microscopically the theory would not be a Palatini theory. It could for instance be a strongly (but not too strongly) coupled chameleon theory (which provided the mass is >> H cosmology would behave like a Palatini theory). These theories do however have 5th forces, and so the possible cosmological evolutions are generally constrained by experiments to be pretty damn close to Lambda CDM.