Probing spin-induced quadrupole moments in precessing compact binaries

Spin-induced quadrupole moments provide an important characterization of compact objects, such as black holes, neutron stars and black hole mimickers inspired by additional fields and/or modified theories of gravity. Black holes in general relativity have a specific spin-induced quadrupole moment, with other objects potentially having differing values. Different values of this quadrupole moment lead to modifications of the spin precession dynamics, and consequently modifications to the inspiral waveform. Based on the spin-dynamics and the associated precessing waveform developed in our previous work, we assess the prospects of measuring spin-induced moments in various black hole, neutron star, and BH mimicker binaries. We focus on binaries in which at least one of the objects is in the mass gap (similar to the $2.6M_{\odot }$ object found in GW190814). We find that for generic precessing binaries, the effect of the spin-induced quadrupole moments on the precession is sensitive to the nature of the mass-gap object, i.e., whether it is a light black hole or a massive neutron star, so that this is a good probe of the nature of these objects. For precessing BH mimicker binaries, this waveform also provides significantly tighter constraints on their spin-induced quadrupole moments than the previous results obtained without incorporating the precession effects of spin-induced quadrupole moments. We apply the waveform to sample events in GWTC catalogs to obtain better constraints on the spin-induced quadrupole moments, and discuss the measurement prospects for events in the O4 run of the LIGO-Virgo-KAGRA Collaboration.

Figure 1

Left: the CDF of each effect for four different cases. PI is an abbreviation for precession induced, AI refers to aligned-spin induced while HA is horizon absorption effect. The Mismatches of C1 and C2 are two order of magnitudes smaller than C3 and C4 because their mass ratios ($q=m_1/m_2>1$) are much greater than C3 and C4. Right: the distribution of fraction of mismatch ($F_i$) defined in Eq. (11) of each effect for four cases. The histogram is normalized for individual effects. We could see the shaded areas which demonstrate the fraction of samples with more than 50% mismatch contribution from the PI effect are about $31\%,23\%,21\%$, and 14%, for cases C1 to C4, respectively.

Figure 2

Sensitivity curves of LIGO-O4, Virgo-O4 [82], ${\mathrm{A}}^{\#}$ [83], and CE [84] detectors.

Figure 3

Left: a corner plot shows the degeneracy between SIQM ${\kappa }_2$ and spins ${\chi }_{1z}$ and ${\chi }_{2z}$ with TaylorF2 waveform model. Right: a comparison between AI, PI, or $\mathrm{PI}+\mathrm{AI}$ effects in precessional waveform models with the same injection while the fractional mismatch of PI is close to AI with $\approx 50\%$ of the total mismatch (Note that there is no PI effect in the TaylorF2 waveform model and we assumed aligned spins in the left plot). The dashed lines in marginalized distributions are the 5% and 95% credible level lines.

Figure 4

Left: posterior distribution of SIQM ${\kappa }_2$, ${\chi }_1$, and ${\chi }_2$ with ${\mathrm{A}}^{\#}$ and CE networks, respectively. Here the range of values above the marginalized distributions are given within a 90% confidence interval. Right: violin plot for $\kappa$ of the $2.6M_{\odot }$ object for the four cases. The violin has been split into two parts: green (left) and orange (right) present constraints from the ${\mathrm{A}}^{\#}$ and CE networks. Each half violin has the same area and the horizontal lines show $1\sigma$ uncertainty bounds.

Figure 5

Posterior distribution of quadrupole moment deviations $\delta {\kappa }_s$, ${\chi }_1$ and ${\chi }_2$ from GW events with the IMRPhenomPv2 waveform model. Left: positive prior on $\delta {\kappa }_s$. The dashed lines in marginalized $\delta {\kappa }_s$ are the 90% upper bound, while the corresponding LVK bounds are depicted by solid lines (they are around 10, very close to each other). Right: a generic prior on $\delta {\kappa }_s$. The result is consistent with the LVK and Krishnendu’s findings [56, 88]. All other dashed lines in the marginalized distributions represent the 5% and 95% credible level lines.

Figure 6

Posterior distribution of quadruposle moment deviations $\delta {\kappa }_s$, ${\chi }_1$, and ${\chi }_2$ with GW151226 (left) and GW191216_213338 (right) with the $\mathrm{IMRPhenomXPHM}+\text{different}$ effect combination. The dashed lines in marginalized $\delta {\kappa }_s$ are 90% upper bound, while they are 5% and 95% credible level lines for ${\chi }_1$ and ${\chi }_2$.

Figure 7

Posterior distribution of individual masses and spins with GW151226 and GW191216_213338. The dashed lines in marginalized plots are 5% and 95% credible level lines. The posterior samples used in this plot are from public available works [89].

Figure 8

This cartoon plot illustrates that a more accurate waveform model does not necessarily imply better constraints on $\delta {\kappa }_s$ (This plot is for illustration purpose only, which is not computed out of an actual calculation. We assume the mismatches reduce to zero as $\delta {\kappa }_s\to 0$, in other words, they reduce to the same waveform models as $\delta {\kappa }_s\to 0$). Three different waveform models are shown; the true waveform in solid blue, and two alternative waveform models in dashed (green is labeled I, and yellow labeled II). For a given mismatch, the true waveform model will be able to constrain $\delta {\kappa }_s^{\star }$ to $[0,\delta {\kappa }_s]$. The incorrect waveform model II will give a poorer constraint, as the range $[0,\delta {\kappa }_s^{\mathrm{II}}]$ is larger than $[0,\delta {\kappa }_s^{\star }]$. However, the incorrect model I will give a tighter constraint.

Figure 9

This plot shows the mismatch as a function of $\delta {\kappa }_s$ for the C5 configuration, which is calculated from actual waveform models using Eq. (16). Three different waveform models are displayed: the most accurate waveform in blue ($\mathrm{IMRPhenomXPHM}+\mathrm{PI}+\mathrm{AI}+\mathrm{HA}$), along with two alternative waveform models in green (IMRPhenomPv2) and orange ($\mathrm{IMRPhenomXPHM}+\mathrm{AI}$). This demonstrates that for C5, the GW151226-like case, the IMRPhenomPv2 model will impose a more stringent constraint on $\delta {\kappa }_s$ for a given mismatch. This aligns with our MCMC simulation results, as depicted in Figs. 5 and 6 for the GW151226 event.

Figure 10

Posterior distribution of quadrupole moment deviations $\delta {\kappa }_s$, ${\chi }_1$, and ${\chi }_2$. Left: result from C5 (GW151226-like) injection with $\mathrm{IMRPhenomXPHM}+\mathrm{AI}$ but recovery with $\mathrm{IMRPhenomXPHM}+\mathrm{AI}$ and IMRPhenomPv2 models. Right: C6 with O4 detector network. The dashed lines in marginalized $\delta {\kappa }_s$ are 90% upper bound, while they are 5% and 95% credible level lines for ${\chi }_1$ and ${\chi }_2$.

Figure 11

Posterior distribution of individual spin polar angles with GW151226 and GW191216_213338. The dashed lines in marginalized plots are 5% and 95% credible level lines. The posterior samples used in this plot are from public available works [89].