Nonminimal gravitational coupling of scalar bound states

We use the large-$N$ approximation to investigate the nonminimal gravitational coupling $\xi R{\phi }^2$ of the scalar bound state in the $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$-symmetric vector model with a ${\varphi }^4$ interaction. If the elementary scalar field is minimally coupled to gravity, $\xi$ is found to assume values in the interval [-1,0] only. For the conformally invariant coupling of the elementary boson, the bound state is always minimally coupled ($\xi =0\mathrm{}$). The induced $\xi$ term is also calculated for spontaneous and explicit symmetry breaking from $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ to O($N-1\mathrm{}$) when the resulting Goldstone bosons can form bound states. The nonminimal gravitational coupling of scalars has important applications in inflationary cosmology.