I-Love-Q in Ho\v{r}ava-Lifshitz Gravity

Ho\v{r}ava-Lifshitz gravity is an alternative theory to general relativity which breaks Lorentz invariance in order to achieve an ultraviolet complete and power-counting renormalizable theory of gravity. In the low energy limit, Ho\v{r}ava-Lifshitz gravity coincides with a vector-tensor theory known as khronometric gravity. The deviation of khronometric gravity from general relativity can be parametrized by three coupling constants: $\alpha$, $\beta$, and $\lambda$. Solar system experiments and gravitational wave observations impose stringent bounds on $\alpha$ and $\beta$, while $\lambda$ is still relatively unconstrained ($\lambda\lesssim 0.01$). In this paper, we study whether one can constrain this remaining parameter with neutron star observations through the universal I-Love-Q relations between the moment of inertia (I), the tidal Love number (Love), and the quadrupole moment (Q), which are insensitive to details in the nuclear matter equation of state. To do so, we perturbatively construct slowly-rotating and weakly tidally-deformed neutron stars in khronometric gravity. We find that the I-Love-Q relations are independent of $\lambda$ in the limit $(\alpha,\beta) \to 0$. Although some components of the field equations depend on $\lambda$, we show through induction and a post-Minkowskian analysis that slowly-rotating neutron stars do not depend on $\lambda$ at all. Tidally deformed neutron stars, on the other hand, are modified in khronometric gravity (though the usual Love number is not modified, as mentioned earlier), and there are potentially new, non-GR Love numbers, though their observability is unclear. These findings indicate that it may be difficult to constrain $\lambda$ with rotating/tidally-deformed neutron stars.