Eccentricity evolution of spinning binaries and its dependence on the equation of state of the components

TL;DR

This work addresses how eccentricity evolves in binaries with spinning components and how the component EOS, via spin-induced quadrupole moments, influences this evolution. By adopting a quasi-Keplerian PN framework and assuming spins perpendicular to the orbital plane, the authors derive an analytical prescription that expresses the instantaneous eccentricity $e_t$ in terms of a reference eccentricity $e_0$ and the PN parameter $y$, expanding up to $ ext{O}(e_0^5)$ and including spin-orbit, spin-spin, and quadrupole contributions. The key contributions include explicit expressions for the PN coefficients $f_1(y)$, $f_3(y)$, and $f_5(y)$, an aligned/antialigned-spin simplification, and a detailed analysis of how the quadrupole moment $Q$ modulates eccentricity evolution across BBH, BNS, and boson-star binaries, as well as EMRIs. The findings show that EOS effects are modest for typical BNSs but can be substantial for boson stars and subsolar-mass NSs, suggesting that eccentricity evolution could become a novel probe of exotic compact objects and formation channels when used in GW data analysis and population studies.

Abstract

We study the evolution of the eccentricity of an eccentric orbit with spinning components. We develop a prescription to express the evolving eccentricity in terms of reference eccentricity and frequency. For that purpose we considered the spins to be perpendicular to the orbital plane. Using this we found an analytical result for the contribution of spin in eccentricity evolution. As a result, we expressed orbital eccentricity in a series of reference eccentricity and gravitational wave frequency. The prescription developed here can easily be used to find arbitrarily higher-order contributions of reference eccentricity. With this we computed the eccentricity upto $\mathcal{O}(e_0^5)$. This result can be used to construct the waveforms of spinning compact objects in an eccentric orbit. Since, our expression depends on the spin induced quadrupole moments, we also study the impact of component properties on the eccentricity evolution through the quadrupole moment. We find for BNSs the evolution depends on the equation of state very mildly unless the NSs are subsolar mass. For subsolar mass NSs the deviations from BH case is comparatively larger and has equation of state dependence. For binary boson stars the deviations are comparatively larger across the mass values. We argue that it may affect our understanding of formation channels and their corresponding populations. We also argue that this can possibly be used as another tool to constrain exoticness of compact objects in a binary.

Figures (9)

• Figure 1: We plot the evolving eccentricity $e_t$ as a function of the post-Newtonian parameter $y$ for BBHs (dashed and dotted black curves) and BNSs with different equations of state (red, green, and blue curves). For all systems, we set $y_0 = 1/\sqrt{6}$ at the ISCO, and evolve the binaries backward in time, i.e., toward lower $y$ values. We set the reference eccentricity as $e_0 = e_t(y_0) = 10^{-4}$. In the left panel, we consider equal-mass components with equal spins. In the right panel, we consider unequal-mass components with equal spins.
• Figure 2: In the above figure we consider equal mass BNS with component masses $.3M_{\odot}$, $.5M_{\odot}$, and $.7M_{\odot}$, with different equations of state (red, green, and blue curves). In the left column we show the eccentricity evolution and in the right column we show evolution of $n$. For all the cases considered $y_0 = 1/\sqrt{6}$ at the ISCO. We set the reference eccentricity as $e_0 = e_t(y_0) = 10^{-4}$. It was shown in Ref. Yagi:2013awa, the low mass NSs has larger QM. Hence, if such low mass NSs do exist and form binaries, the EoS impact will be larger on the orbital dynamics.
• Figure 3: We plot the evolving eccentricity $e_t$ as a function of the post-Newtonian parameter $y$ for BBHs (dashed and dotted black curves) and binary boson stars with different QMs. Red, green, and blue curves represent $Q=60,\, 25, \, 5$. For all systems, we set $y_0 = 1/\sqrt{6}$ at the ISCO, and evolve the binaries backward in time, i.e., toward lower $y$ values. We set the reference eccentricity as $e_0 = e_t(y_0) = 10^{-4}$. We consider equal-mass components with equal spins.
• Figure 4: In the above plot we show the evolution of $a$ and $n$, defined in Eq. \ref{['eq:orbital-quantities']}. Red, green, and blue curves represent $Q=60,\, 25, \, 5$. $a$ and $n$ deviates from BH values for larger $Q$ and $\chi$. The deviations are stronger only for larger $y$.
• Figure 5: In the above figure we consider an extreme mass ratio inspiral with $\nu=10^{-4}$, with the primary as a boson star of QM $Q=15$, $e_0 =10^{-4}$ at $y_0=1/\sqrt{6}$. In the right figure of the first row we show eccentricity evolution. The eccentricity evolution shows significant EoS dependence. The nonspining binary with same configuration is shown in cyan colour. With decreasing spin the eccentricity values decreases to go below nonspining value before increasing to reach nonspining value. The inset zooms in on the overlapping curves of slowly spinning configuration to demonstrate the QM-spin competition for low spins. In the rest of the panels we show the evolution of $a$ and $n$ defined in Eq. \ref{['eq:orbital-quantities']}. $a$ and $n$ deviates from non-spinning configuration for larger $Q$ and $\chi$. The deviations are stronger only for larger $y$ as found in other figures also. Clearly EoS dependent deviation is observable in all of the plots.