beyond $2\,\rho_0$. Therefore, assuming continuity an extrapolation of $U_\Lambda$ to high denstites is supposed to be well justified using the following ansatz for $\rho > \rho_0$: $$U_\Lambda(\rho) = u_0 + u_1\left({\rho\over\rho_0}-1\right) + u_2\left({\rho\over\rho_0}-1\right)^2\,. 
\label{eq:extrap}$$ The parameters $u_1$ and $u_2$ are determined by fits to the calculated single-particle potentials at densities below $\rho_c$. With $u_0$ fixed at $\rho=\rho_0$, it turns out that the high-density behaviour is indeed governed by the quadratic term, with $u_2 \gg u_1$.

![image](ULambda)

![image](NStarCP)

Results for the high-density extrapolation of $U_\Lambda$ in symmetric nuclear matter and neutron matter are displayed in fig.\[fig:ulambda\]. With two-body $YN$ interactions only, the linearly increasing $U_\Lambda$ at densities $\rho\gtrsim 2\rho_0$ is reminiscent of a Hartree potential. When increasingly repulsive three-body interactions are added, the leading $\rho^2$ dependence of the 3BF takes over at high densities. Note that in neutron matter, the strong repulsion in $U_\Lambda$ exceeds 100 MeV already at $\rho \simeq 3.5 \,\rho_0$ and reaches $U_\Lambda > 300$ MeV at $\rho\simeq 5\,\rho_0$, the density range characteristic of the inner core of neutron stars.

Hyperon and neutron chemical potentials in neutron stars
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If the chemical potentials of the $\Lambda$ hyperon and neutron fulfill the condition $\mu_\Lambda = \mu_n$ at some density 