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.
Categories: gr-qc astro-ph.CO hep-ph hep-th