Primordial Nucleosynthesis in Conformal Weyl Gravity

Recently conformal Weyl gravity has been considered as a candidate alternative gravity theory. This fourth-order theory is attractive because it is the only metric theory of gravity which is invariant under local conformal transformations of the metric. We calculate the primordial light element abundances in this theory. The major difference {}from the standard cosmology is that the universe expands far more slowly throughout the nucleosynthesis epoch. The production of $^4\rm{He}$ depends strongly on $\eta$, the ratio of baryons to photons. For $\eta = 10^{-8}$ the mass fraction of $^4{\rm He}$ is $X_4 \simeq 0.25$ and the number densities relative to hydrogen for $^2{\rm H}$, $^3{\rm He}$ and $^7{\rm Li}$ are $n(^2{\rm H})/n({\rm H}) \simeq 9\times 10^{-20}$, $n(^3{\rm He})/n({\rm H})\simeq 4 \times 10^{-18}$ and $n(^7{\rm Li})/n({\rm H}) \simeq 10^{-13}$. This value of $\eta$ corresponds to a baryon mass density close to the standard model critical density. However, adjusting $\eta$ to give a reasonable helium yield forces the deuterium and lithium yields to be small enough that the theory cannot be reconciled with observations.