Appearances are deceptive: Can graviton have a mass?

We study the dynamics of linear gravitational perturbations on cosmological backgrounds of massive fermionic fields. We observe that, when gravitational and matter action are expanded to quadratic order in gravitational perturbations on cosmological backgrounds, the graviton appears to have an off-shell mass. We derive a consistent set of two equations for the evolution of linear classical and quantum gravitational perturbations on general cosmological backgrounds, and demonstrate that the graviton mass disappears at the level of equations of motion (on-shell). In the case we consider the expansion of the Universe is driven by the one-loop backreaction of fermions, and the dynamical gravitons evolve on the same background. These equations govern the evolution of linear gravitational perturbations on general cosmological matter backgrounds. A concrete one-loop calculation is performed for the simple case of massive Dirac fermions when the temperature of the cosmological fluid changes adiabatically when compared with the expansion rate of the Universe.

💡 Research Summary

The paper tackles the long‑standing question of whether the graviton can acquire a mass when quantum matter fields drive the expansion of the Universe. The authors consider a spatially flat Friedmann‑Lemaître‑Robertson‑Walker (FLRW) background in D dimensions and introduce a massive Dirac fermion field whose mass may have both scalar ( (m_R) ) and pseudo‑scalar ( (m_I) ) components. By performing a Weyl rescaling, the background metric becomes conformally flat, (g_{\mu\nu}=a^2(\eta)(\eta_{\mu\nu}+\kappa h_{\mu\nu})), where (\kappa^2=16\pi G) is the loop‑counting parameter and (h_{\mu\nu}) denotes the graviton perturbation.

The authors expand both the Hilbert‑Einstein action and the Dirac action to second order in (\kappa). The quadratic gravitational action (S^{(2)}{\text{HE}}) contains the usual kinetic terms for the graviton as well as non‑derivative pieces proportional to (h^2) and (h{\mu\nu}h^{\mu\nu}). These non‑derivative terms look like a mass term for the graviton, with an effective “off‑shell” mass squared of order ((D-2)H^2) (where (H) is the conformal Hubble rate).

To treat the fermionic sector consistently, the paper adopts the two‑particle‑irreducible (2PI) formalism. The fermionic two‑point functions (S_{ab}(x;x’)) (with Keldysh indices (a,b=\pm)) are introduced, and the one‑loop contribution (\Gamma^{(1)}{\text{D}}=\frac{i\hbar}{2}\mathrm{Tr}\ln S) is added to the effective action. The coincident limit of the fermionic kinetic kernel (K{\mu\nu}(x;x)) and the scalar kernel (L(x;x)) can be expressed in terms of the background energy‑momentum tensor: \