We use the mostly positive metric signature. We contract tensor indices with the metric $g_{\mu \nu }$, whose determinant is $g$ and Ricci scalar $R$. Finally, we use natural units, and $M_{\mathrm{Pl}}$ denotes the reduced Planck mass.

The action of quadratic gravity can generically be written as such up to boundary and topological terms (see, e.g., Ref. [9]).

The natural possibility of a “creation out of nothing” is a compelling one here, but it certainly deserves further investigation. In particular, finding proper instanton solutions and analyzing the path integral of QQG would constitute interesting follow-up work.

One may question how large $\mathcal{N}$ can reasonably be. In theoretical settings, such as holography, $\mathcal{N}$ is often taken to be arbitrarily large, to ensure perturbative gravity in the bulk. Here, we remain agnostic about whether or not a very large $\mathcal{N}$ represents a fine-tuning problem. We should further note that these matter fields would probably end up being confined to the UV, together with all the degrees of freedom of QQG, in a similar fashion to quarks and gluons in QCD.

Here, we ignore the impact of the running of the coupling to Gauss-Bonnet’s invariant $\mathcal{G}$, which is not widely studied due to its topological nature. However, this may affect cosmological evolution—as in $f(R,\mathcal{G})$ theories, which are widely studied in the literature, e.g., [66]—but we defer that consideration to future work.

Note that during the slow-roll phase of inflation, due to the near de Sitter nature of spacetime, different choices of curvature invariants for the RG scale are generically equivalent to the Ricci scalar. However, they will start to differ as we approach the end of inflation, and thus, they could lead to different reheating mechanisms in the strongly coupled regime.

The contribution from the Weyl tensor vanishes on a homogeneous and isotropic background. This is not true once cosmological perturbations are introduced. Still, contributions from the Weyl tensor to, e.g., tensor perturbations are usually highly suppressed [34], and we do not expect this result to be affected by small logarithmic running.

Cold dark matter with a cosmological constant ($\mathrm{\Lambda }\mathrm{CDM}$) refers to the standard model of cosmology, while the Chevallier-Polarski-Linder [83, 84] parametrization for dynamical dark energy (often called $w_0{w}_a\mathrm{CDM}$) stands as an alternative where the dark-energy equation of state depends linearly on the cosmological scale factor. The recent BAO data of DESI point toward a slight preference for the latter [80].