Remarks on the matter-graviton coupling

We point out a generic inconsistency of the coupling of ordinary gravity as described by general relativity with matter invariant only under unimodular diffeomorphisms, and some previously studied exceptions are pointed out. The most general Lagrangian invariant under unimodular diffeomorphism up to dimension-five operators is determined, and consistency with existing observations is studied in some cases.

Figure 1

The shadowed region shows experimentally allowed values for $|m_2^2|$ (in $m^{-2}$) and ${\mu }_2$ for given values of $c_2$ and $c_3$, expressed in terms of $\Delta c$, and in the range $\lambda \in {10}^{-9}\mathrm{m}–{10}^{-6}\mathrm{m}$.

Figure 2

Experimentally allowed region in the plane ($c_2$, $c_3$), for a couple of values of the mass, in the range $\lambda \in {10}^{-6}\mathrm{m}–{10}^{-2}\mathrm{m}$. The plot is restricted to the zone where the curve appreciably deviates from the parabola $\Delta c=0$.

Figure 3

Experimentally allowed region in the plane ($c_2$, $c_3$), for a couple of values of the mass, in the range $\lambda \in {10}^{-2}\mathrm{m}–{10}^{14}\mathrm{m}$. We only show the positive $c_2$ branch. The parabola is indistinguishable from the $c_2$ axis.

Figure 4

Experimentally allowed region in the plane ($c_2$, $c_3$) and in the range $\lambda \in {10}^{-2}\mathrm{m}–{10}^{14}\mathrm{m}$ for a very tiny mass. That allows us to see the parabola, which was hidden in the previous figure. We only show the zone closest to the $c_3$ axis.