A Possible Solution to the Helium Anomaly of EMPRESS VIII by Cuscuton Gravity Theory

We discuss cosmology based on the cuscuton gravity theory to resolve the anomaly of the observational $^4$He abundance reported by the EMPRESS collaboration. We find that the gravitational constant $G_{\rm cos}$ in Friedmann equation should be smaller than the Newton's constant ${G_{\rm N}}$ such that ${\Delta G_{\rm N}}/{G_{\rm N}} \equiv (G_{\rm cos}-G_{\rm N})/{G_{\rm N}} = -0.085_{-0.028}^{+0.026} \quad(68 \% \text { C.L. })$ in terms of big-bang nucleosynthesis, which excludes ${\Delta G_{\rm N}}=0$ at more than 95~$\% \text { C.L. }$ To fit the data, we obtain a negative mass squared of a non-dynamical scalar field with the Planck-mass scale as $\sim - {\mathcal{O}}(1) {M_{\rm PL}^2} ({\mu}/{0.5 M_{\rm PL}})^{4}$ with the cuscuton mass parameter $\mu$. This fact could suggest the need for modified gravity theories such as the cuscuton gravity theory with a quadratic potential, which can be regarded as the low-energy Ho\v{r}ava-Lifshitz gravity and might give a hint of quantum gravity.