Measurability of the tidal polarizability of neutron stars in late-inspiral gravitational-wave signals

The gravitational wave signal from a binary neutron star inspiral contains information on the nuclear equation of state. This information is contained in a combination of the tidal polarizability parameters of the two neutron stars and is clearest in the late inspiral, just before merger. We use the recently defined tidal extension of the effective one-body formalism to construct a controlled analytical description of the frequency-domain phasing of neutron star inspirals up to merger. Exploiting this analytical description we find that the tidal polarizability parameters of neutron stars can be measured by the advanced LIGO-Virgo detector network from gravitational wave signals having a reasonable signal-to-noise ratio of $\rho =16$. This measurability result seems to hold for all the nuclear equations of state leading to a maximum mass larger than $1.97M_{\odot }$. We also propose a promising new way of extracting information on the nuclear equation of state from a coherent analysis of an ensemble of gravitational wave observations of separate binary merger events.

Figure 1

Successive PN approximants to the (SPA) tidal contribution to the EOB phase as function of the $\ell =m=2$ GW frequency. The leftmost vertical line indicates 450 Hz for a $1.4M_{\odot }+1.4M_{\odot }$ BNS system. The rightmost vertical line indicates the contact frequency that is taken as a fiducial analytical definition of the moment of merger. The plot refers to a $\mathcal{C}=0.16$, $\gamma =2$ polytropic model.

Figure 2

Difference between the 2.5PN-expanded tidal (Fourier) phase and the corresponding exact EOB one obtained by integrating Eq. (20). Each curve refers to a $\gamma =2$ polytropic model with different compactness. The vertical lines indicate the corresponding contact frequency.

Figure 3

Integrands, per frequency octave, of the integrals determining the measurability of $\mathcal{M}$, $\nu$, $\rho$ (SNR) and ${\lambda }_T$. While most of the SNR is gathered around frequencies $\widehat{f}=f/(56.56\mathrm{Hz})\sim 1$, the measurability of $\mathcal{M}$ and $\nu$ is concentrated towards lower frequencies ($\widehat{f}=f/f_0<1$), and that of the tidal parameter ${\lambda }_T$ gets its largest contribution from the late inspiral up to the merger. The rightmost vertical line indicates the merger frequency for $\mathcal{C}=0.1645$, while the leftmost vertical line marks 450 Hz for a $1.4M_{\odot }+1.4M_{\odot }$ BNS system.

Figure 4

Measurability of the tidal polarizability parameter $G{\mu }_2$ (in units of ${\mathrm{km}}^5$) as a function of the neutron star mass for a sample of realistic EOS from Table . This plot refers to the observation (at the SNR level $\rho =16$) of the gravitational wave signal from an equal-mass BNS merger as seen by a single advanced LIGO detector. The solid lines represent the values of $G{\mu }_2$ as a function of the NS mass, while the dashed lines represent the $1\sigma$ (68% confidence level) expected statistical errors. The vertical line marks the canonical NS mass $1.4M_{\odot }$. Note that over a wide range of masses each solid line lies comfortably above the corresponding measurability threshold, therefore indicating that the advanced LIGO-Virgo detector network can significantly measure $G{\mu }_2$.

Figure 5

Difference between the full SPA tidal phase ${\Psi }_{k_{2,3,4}}^T$ obtained from a EOB waveform computed with an $A^{\mathrm{tidal}}$ including $k_2$, $k_3$ and $k_4$, and the tidal phase computed with $k_2$ only. The figure refers to a $\gamma =2$, rest-mass polytrope BNS model with $\mathcal{C}=0.16$ and ${\overline{\alpha }}_2^{(2)}=6$. The vertical dashed line indicates the contact frequency.

Figure 6

Approximate linearity of ${\Psi }_{\mathrm{EOB}}^T$ with respect to ${\kappa }_2^T$, with ${\overline{\alpha }}_2^{(2)}=6$. Three equal-mass (polytropic) BNS systems of compactnesses $\mathcal{C}=\{0.14,0.16,0.18\}$ are compared. The vertical dashed lines indicate the corresponding contact frequencies.