Cosmological Perturbation Theory in $f(Q,T)$ Gravity

Abstract

We developed the cosmological linear theory of perturbations for $f(Q,T)$ gravity, which is an extension of symmetric teleparallel gravity, with $Q$ the non-metricity and $T$ the trace of the stress-energy tensor. By considering an ansatz of $f(Q,T)=f_1(Q)+f_2(T)$, which has been broadly studied in the literature and the coincident gauge where the connection vanishes, we got equations consistent with $f(Q)$ gravity when $f_{T}=0$. In the case of the tensor perturbations, the propagation of gravitational waves was found to be identical to $f(Q)$, as expected. For scalar perturbations, outside the limit $f_T = 0$, we got that the coupling between $Q$ and $T$ in the Lagrangian produces a coupling between the perturbation of the density and the pressure. The presence of $T$ in the Lagrangian breaks the equation of the conservation of energy, which in turn breaks the standard $\rho' + 3\mathcal{H} (\rho+p) = 0$ relation. We also derived a coupled system of differential equations between $\delta$, the density contrast and $v$ in the $\mathcal{H}<<k$ limit and with negligible time derivative of the scalar perturbation potentials, which will be useful in future studies to see whether this class of theories constitute a good alternative to dark matter. These results might also enable to test $f(Q,T)$ gravity with CMB and standard siren data that will help to determine if these models can reduce the Hubble constant tension and if they can constitute an alternative to the $\Lambda$CDM model.