Parametrized tests of general relativity using eccentric compact binaries

Astrophysical population simulations predict that a subset of dynamically formed binary black holes (BBHs) may possess eccentricity $\ensuremath{\gtrsim}0.1$ at a gravitational wave (GW) frequency of 10 Hz. Presently, the LIGO-Virgo-KAGRA (LVK) Collaboration tests general relativity (GR) assuming that the binary eccentricity has decayed well before it enters the detector's frequency band. The detection of an eccentric binary could allow us to test GR in a regime inaccessible with binaries in circular orbits. Previous works have also shown that binary eccentricity can bias GR tests if unaccounted for. Here we develop two methods to extend parametrized tests of GR to eccentric binaries. The first method extends the standard null parametrized test for quasicircular binaries by adding fractional deviations at each post-Newtonian (PN) order in the eccentric part of the GW phasing (assuming the small-eccentricity limit). Simultaneous measurement of the circular and eccentric deviation parameters ($\ensuremath{\delta}\stackrel{^}{\ensuremath{\varphi}},\ensuremath{\delta}{\stackrel{^}{\ensuremath{\varphi}}}^{e}$) allows us to constrain deviations from GR for eccentric binaries. While strong constraints on the deviation parameters are not achievable with LIGO's projected sensitivity, Cosmic Explorer (CE) can constrain 1PN deviations to $|\ensuremath{\delta}{\stackrel{^}{\ensuremath{\varphi}}}_{2}|\ensuremath{\lesssim}2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$ and $|\ensuremath{\delta}{\stackrel{^}{\ensuremath{\varphi}}}_{2}^{e}|\ensuremath{\lesssim}{10}^{\ensuremath{-}1}$ for spinning BBHs. The multibanding of LISA and CE observations can constrain these deviations to $|\ensuremath{\delta}{\stackrel{^}{\ensuremath{\varphi}}}_{2}|\ensuremath{\lesssim}3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ and $|\ensuremath{\delta}{\stackrel{^}{\ensuremath{\varphi}}}_{2}^{e}|\ensuremath{\lesssim}2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$. The second method looks for GR deviations in the rate of periastron advance. We modify the relation between the radial and azimuthal frequencies by modifying the periastron advance per orbit $\mathrm{\ensuremath{\Delta}}\mathrm{\ensuremath{\Phi}}\ensuremath{\equiv}2\ensuremath{\pi}k$ via $k\ensuremath{\rightarrow}k(1+\mathrm{\ensuremath{\Delta}}\ensuremath{\alpha})$. We derive a parametrized phase expansion for nonspinning eccentric binaries in terms of a non-GR deviation parameter $\mathrm{\ensuremath{\Delta}}\ensuremath{\alpha}$ that captures deviations in the periastron advance relative to the GR prediction ($\mathrm{\ensuremath{\Delta}}{\ensuremath{\alpha}}^{\mathrm{GR}}\ensuremath{\rightarrow}0$). The parameter $\mathrm{\ensuremath{\Delta}}\ensuremath{\alpha}$ can be constrained with LIGO to $|\mathrm{\ensuremath{\Delta}}\ensuremath{\alpha}|\ensuremath{\lesssim}4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$ (with $1\ensuremath{\sigma}$ confidence). For a typical binary neutron star (BNS) system, $\mathrm{\ensuremath{\Delta}}\ensuremath{\alpha}$ can be measured with $|\mathrm{\ensuremath{\Delta}}\ensuremath{\alpha}|\ensuremath{\lesssim}8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ in LIGO. With CE sensitivity, the bounds on $\mathrm{\ensuremath{\Delta}}\ensuremath{\alpha}$ improve by $\ensuremath{\sim}1$ order of magnitude for BBHs and $\ensuremath{\sim}2$ orders of magnitude for BNS. Multiband sources observed by LISA and CE provide an improved constraint of $|\mathrm{\ensuremath{\Delta}}\ensuremath{\alpha}|\ensuremath{\lesssim}3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$. The space-based detector DECIGO provides the best constraint on $\mathrm{\ensuremath{\Delta}}\ensuremath{\alpha}$ with $|\mathrm{\ensuremath{\Delta}}\ensuremath{\alpha}|\ensuremath{\lesssim}8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$.