Environmental effects in stellar mass gravitational wave sources II: Joint detections of eccentricity and phase shifts in binary sub-populations

TL;DR

This work shows that moderately eccentric stellar-mass GW sources can dramatically boost the detectability of environmental dephasing (EE) in GW signals by leveraging high-order harmonics. Using a fast, Newtonian frequency-domain eccentric waveform and EE dephasing prescriptions, the authors quantify how the EE delta-SNR scales as $\ell_{\max}^{1-n}$ and how higher harmonics sample earlier binary evolution, enhancing phase shifts. They demonstrate, across LVK, CE, and ET sensitivities, that eccentric tails of the population can render EE detectable even for weaker environmental effects, with the strongest gains for next-generation detectors and larger $\ell_{\max}$. The results imply that joint inference of eccentricity and EE can probe formation channels (dynamical, AGN) and environmental conditions (Roemer delays, gas torques) in current and upcoming GW catalogs, significantly advancing GW astrophysics and multi-messenger inference.

Abstract

We demonstrate that the properties of eccentric gravitational wave (GW) signals enhance the detectability of GW phase shifts caused by environmental effects (EEs): The signal-to-noise ratio (SNR) of EEs can be boosted by up to $\ell_{\rm max}^{1 - n}$ with respect to corresponding circular signals, where $\ell_{\rm max}$ is the highest modeled eccentric GW harmonic and $n$ is the frequency scaling of the GW dephasing prescription associated to the EE. We investigate the impact on a population level, adopting plausible eccentricity distributions for binary sources observed by LIGO/Virgo/Kagra (A+ and A\# sensitivities), as well as Cosmic Explorer (CE) and the Einstein Telescope (ET). For sources in the high eccentricity tail of a distribution ($e \gtrsim 0.2$ at 10 Hz), phase shifts can systematically be up to $\ell_{\rm max}^{1 - n}$ times smaller than in a corresponding circular signal and still be detectable. For typical EEs, such as Roemer delays and gas drag, this effect amounts to SNR enhancements that range from $10^2$ up to $10^5$. For CE and ET, our analysis shows that EEs will be an ubiquitous feature in the eccentric tail of merging binaries, regardless of the specific details of the formation channel. Additionally, we find that the joint analysis of eccentricity and phase shift is already plausible in current catalogs if a fraction of binaries merge in AGN migration traps.

Figures (5)

• Figure 1: Illustration of the fundamental insight motivating this work. Eccentric binaries radiate GW at the characteristic frequency associated to pericenter passage. Therefore, they produce GW at high frequencies (in particular at the characteristic "peak" frequency) over a larger portion of their entire evolution. They enter the sensitive frequency band of ground based GW detectors at a wider separation, at which environmental effects are strong with respect to relativistic effects. Therefore, we expect a stronger trace of binary environments in the GW signal of eccentric sources.
• Figure 2: Top panel: Characteristic strain tracks (given by $\tilde{h}(f)\times f$) of a 8 M$_{\odot}$ + 8 M$_{\odot}$ binary at $z=0.5$ compared to various detector sensitivity curves (dashed lines), for some sample eccentricities at 10 Hz (here meaning when the GW $\ell =2$ harmonic reaches 10 Hz). The solid lines represent the GW envelope, while the thin lines show the interference of the various harmonics. The tracks are truncated at the binary innermost stable circular orbit. Bottom panel: residual strain between a the vacuum waveform and the corresponding waveform with a phase shift of $10^{-3}$ radians at 10 Hz and a frequency scaling of $f^{-13/3}$. Note how the residuals are shifted to higher frequencies as the eccentricity increases, entering the band of GW detectors. These particular waveforms are computed with a maximum of 50 harmonics.
• Figure 3: Contour plots for the $\delta$SNR for a binary source of GW with $m_1 = m_2 =$ 8 M$_{\odot}$ located at a typical redshift for the given detector configuration. The contours are computed for a dephasing prescription with $n=-13/3$, and show the dependance of the $\delta$SNR on the magnitude of the dephasing and the eccentricity at 10 Hz. The top row is computed for $\ell_{\rm max}=10$, while the bottom row is for $\ell_{\rm max}=50$. Note how the detectability of the dephasing is greatly increased as soon as significant power is distributed in the higher harmonics of the GW emission, i.e. for eccentricities of $\gtrsim 0.1$. The phase space regions with approximately constant $\delta$SNR as a function of $A_2^{10\rm Hz}$ result from the saturation of the eccentric harmonics (see text).
• Figure 4: Increases in the $\delta$SNR as a function of the residual eccentricity at 10 Hz with respect to a circular signal. The results are computed for different detectors (coloured lines), dephasing power laws (panels, see Eq. \ref{['eq:dephfamI']}) and for two representative choices for $\ell_{\rm max}$ (solid and dashed lines). The curves are computed for a dephasing amplitude of $A_2^{10 \rm Hz} =10^{-15}$, ensuring that no eccentric harmonic is ever saturated. Note that the maximum achieved $\delta$SNR factors scales roughly as $\ell_{\rm max}^{1 -n}$, and that these values are reached for a range of moderate eccentricities. Note also that the $n=-5/3$ results are essentially showing an increased capacity to determine the binary's chirp mass (recall the $F^{-5/3}$ scaling of the vacuum GW phase), a phenomenon already studied in 2020moore.
• Figure 5: Contours for the fraction $\epsilon$ of GW signals within the tail of a realistic eccentricity distribution (see text), that have a phase shift with $\delta$SNR>3, here for an example dephasing with $n=-13/3$. The high eccentricity tail is defined by a cut--off value $e_{\rm cut}$ and the results are computed as a function of the dephasing amplitude $A_2^{10 \rm Hz}$, here for a representative LVK and CE/ET source consisting of a 8 M$_{\odot}$ + 8 M$_{\odot}$ binary placed at $z=0.2$ and $z=3$, respectively. Note how focusing on the high eccentricity sources increases the chances of detecting weaker EEs. The binary cumulative distribution (CDF) as a function of $e_{\rm cut}$ is over-plotted on the contours to highlight the trade-off between the quantity of sources with $e_{10 \rm Hz}> e_{\rm cut}$ and the detectability boost for EE in sources with high eccentricity.