# Uncertainty limits on neutron star radius measurements with gravitational
waves

Katerina Chatziioannou  Department of Physics, California Institute of
Technology, Pasadena, California 91125, USA  LIGO Laboratory, California
Institute of Technology, Pasadena, California 91125, USA

###### Abstract

Upcoming observing campaigns with improved detectors will yield numerous
detections of gravitational waves from neutron star binary inspirals. Rare
loud signals together with numerous signals of moderate strength promise
stringent constraints on the properties of neutron star matter, with a
projected radius statistical uncertainty of $50-200$m with ${\cal{O}}(2000)$
sources. Given this precision we revisit all analysis assumptions and identify
sources of systematic errors, quantify their impact on radius extraction, and
discuss their relative importance and ways to mitigate them.

##  I Introduction

Astronomical observations constrain the macroscopic properties of neutron
stars (NSs) and the behavior of dense, cold matter Lattimer and Prakash
(2016); Oertel  _et al._ (2017); Baym  _et al._ (2018). Among them,
gravitational wave (GW) observations of binary NS (BNS) inspirals lead to a
measurement of the masses and tidal properties of the stars Chatziioannou
(2020); Dietrich  _et al._ (2021). The binary masses affect the evolution of
the early, minute-long signal, while each NS’s response to its companion’s
gravitational tidal field leaves an imprint on the signal during the final
coalescence stages.

The mutual tidal interaction induces a quadrupole moment on each NS which
accelerates the GW inspiral. The leading-order effect is quantified through
the tidal deformability $\Lambda\equiv(2/3)k_{2}(R/m)^{5}$ Flanagan and
Hinderer (2008); Hinderer  _et al._ (2010), where $k_{2}$ is the Love number
of a NS with mass $m$ and radius $R$. Measuring $\Lambda$ offers complementary
information to traditional radius measurements through $k_{2}$, while the
strong mass dependence implies tighter constraints for less massive, and thus
more deformable, NSs. The first BNS coalescence detected with GWs, GW170817
Abbott  _et al._ (2017a), had a signal-to-noise ratio (SNR) of $\rho=32$, and
a combined tidal deformability $\tilde{\Lambda}$ Favata (2014); Wade  _et al._
(2014) of $300^{+420}_{-230}$ at the 90% credible level Abbott  _et al._
(2018a, 2019a). This result has been extensively shown to disfavor very stiff
nuclear matter and large NS radii and tidal deformabilities Annala  _et al._
(2018); Raithel  _et al._ (2018); Radice and Dai (2019); Coughlin  _et al._
(2019); Essick  _et al._ (2020a); Landry and Essick (2019); De  _et al._
(2018); Raaijmakers  _et al._ (2020); Capano  _et al._ (2020); Dietrich  _et
al._ (2020); Landry  _et al._ (2020); Essick  _et al._ (2020b); Abbott  _et
al._ (2020a) and to be in agreement with further astronomical and terrestrial
constraints Landry  _et al._ (2020); Legred  _et al._ (2021); Abbott  _et al._
(2020a); Raaijmakers  _et al._ (2021); Essick  _et al._ (2020b, 2021); Biswas
(2021); Biswas  _et al._ (2021); Miller  _et al._ (2021); Al-Mamun  _et al._
(2021); Reed  _et al._ (2021); Tews  _et al._ (2019a, b).

Scheduled or planned GW detector upgrades and observing campaigns Abbott  _et
al._ (2013) are expected to yield further BNS detections and improve on the
overall constraints by both combining information from multiple events of
different masses Lackey and Wade (2015); Abdelsalhin  _et al._ (2018);
Hernandez Vivanco  _et al._ (2019); Landry  _et al._ (2020) and detecting
louder signals. The uncertainty in tidal parameters scales as $\rho^{-1}$ Wade
_et al._ (2014), however, the tidal deformability affects the GW signal
primarily during the late stages of the coalescence for frequencies $\gtrsim
400$Hz. Lower frequencies are essential for identifying the signal and
estimating the masses, but tidal inference primarily relies on the SNR
accumulated, and thus the expected detector performance, at this frequency
range.

Since the SNR is inversely proportional to the detector sensitivity, we can
make rough estimates about the expected constraints on the NS tidal
deformability. The LIGO-Livingston Aasi  _et al._ (2015) strain sensitivity in
the relevant frequencies improved by a factor of $\sim$2 between the second
Properties of the binary neutron star merger GW170817 ; LIGO Scientific
Collaboration, Virgo Collaboration (2019) and third observing runs Abbott  _et
al._ (2020b, c). Design sensitivity could bring another factor of $\sim$1.5
Barsotti  _et al._ (2018), with the A+ and Voyager Adhikari  _et al._ (2020);
Evans  _et al._ (2018) upgrades yielding improvements by factors of $\sim$2
and $\sim$1.5 respectively. Next generation 3G detectors are envisioned to
have $\sim$10 times better strain sensitivity than advanced LIGO Abbott  _et
al._ (2017b); Reitze  _et al._ (2019a, b); Hild  _et al._ (2011); Punturo  _et
al._ (2010). Though these improvements are not uniform across the frequency
band, they roughly suggest that a GW170817-like event would have an SNR
($\tilde{\Lambda}$ 90% uncertainty) of $100(200)$ at design sensitivity,
$200(100)$ with A+, $300(66)$ with Voyager, and $1000(20)$ with 3G detectors.

The total number of sources observed per detector upgrade depends on the BNS
merger rate and its distribution with redshift with a current estimate of
$320^{+490}_{-240}\mathrm{Gpc}^{-3}\mathrm{yr}^{-1}$ mergers uniformly
distributed in comoving volume and for uniform NS masses in $(1,2.5)M_{\odot}$
Abbott  _et al._ (2021). With these specifications, the median merger rate,
and a detection SNR threshold of $12$ we use the GW-Toolbox Yi  _et al._
(2021) to estimate ${\cal{O}}(10)$ detections per year at design sensitivity,
${\cal{O}}(100)$ for A+, and ${\cal{O}}(500)$ for Voyager. The latter is $7$
times lower than the estimate of Adhikari  _et al._ (2020), likely due to
differences in the merger rate and SNR threshold. Predictions for 3G detectors
are subject to the uncertain redshift distribution of mergers, but estimates
suggest ${\cal{O}}(1000)$ sources per day Regimbau  _et al._ (2012); Sachdev
_et al._ (2020).

We estimate the total SNR accumulated by these detections by adopting a
$\rho^{-4}$ distribution Chen and Holz (2014), appropriate for detectors up to
Voyager that get the majority of their sources from redshifts up to $0.3$
Adhikari  _et al._ (2020). We simulate $100$ source catalogs and compute the
total SNR by summing the per-event SNRs in quadrature to find a median total
SNR of 60(200)[450]{650} with 10(100)[500]{1000} sources. The total SNR from
3G detectors is expected to be ${\cal{O}}(10^{4})$ Haster  _et al._ (2020).
These estimates are conservative if dedicated high-frequency detectors join
the network Bailes  _et al._ (2019); Ganapathy  _et al._ (2021).

The corresponding NS radius accuracy is shown in Fig. 1 for different NS
masses and radii, assuming we can perfectly convert a ($m,\tilde{\Lambda}$)
measurement to $R$. The $\Lambda\sim(R/m)^{5}$ scaling results in increased
radius uncertainty with higher/lower NS mass/radius due to intrinsically
weaker binary tidal interactions. An SNR of $1000$, achieved either by
observing a GW170817-like event with 3G detectors or with $4$ years of Voyager
operation, would result in a measurement of $\tilde{\Lambda}$ to $20$ and $R$
to $50-200$m for different NS masses. A more moderate total SNR of $200$ from
$\sim$100 sources would lead to a $\tilde{\Lambda}$ ($R$) uncertainty of $100$
($1$km) at $1.6M_{\odot}$, consistent with the more detailed simulations of
Landry  _et al._ (2020). The final constraint achieved on NS radii will be a
combination of these per-mass estimates depending on the astrophysical NS mass
distribution.

Figure 1: Radius 90% uncertainty as a function of the uncertainty in the
combined tidal deformability for different NS masses and radii. The top axis
gives the corresponding SNR, achieved with either a single source or a
combination of sources. Estimates are based on GW170817’s
$\Delta\tilde{\Lambda}\sim 650$ at SNR $32$. Solid (dot-dashed) vertical lines
are projected results from a GW170817-like event (a catalog of BNSs) with
different detectors and observing durations.

The above are not detailed predictions about the expected NS constraints from
future detectors; such estimates would require a precise treatment of -among
others- the BNS merger rate, its redshift distribution, the NS mass
distribution, the broadband detector performance, the network duty cycle, etc.
However, they provide a projection that GWs could result in a $\sim$100m
radius measurement within the decade with Voyager and beyond with 3G
detectors. This radius constraint would also improve if effects such as
dynamical tides Pratten  _et al._ (2020); Schmidt and Hinderer (2019); Ma  _et
al._ (2021) are detected as they are qualitatively different than the standard
adiabatic tides considered here and not captured by the $\rho^{-1}$ scaling.
Reaching this projected precision relies on ascertaining that every aspect of
the GW analysis induces potential systematic errors that are fully quantified
and brought below statistical uncertainties.

##  II Gravitational wave analysis

Analysis of GW data $d$ to extract source parameters $\theta$ relies on
modeling the signal with a waveform template $h(\theta)$ under some model for
the detector noise. The likelihood function in the frequency domain is Veitch
_et al._ (2015); Romano and Cornish (2017)

$\mathrm{log}L\sim-\frac{1}{2}(d-h(\theta)|d-h(\theta)),$ (1)

with the noise-weighted inner product

$(a|b)\equiv 2\int\frac{a^{*}(f)b(f)+b^{*}(f)a(f)}{S_{n}(f)}df,$ (2)

where an asterisk denotes complex conjugation and $S_{n}(f)$ is the power
spectral density (PSD) of the noise. The likelihood and a prior for $\theta$
give the posterior probability.

The above allows us to identify the ingredients of parameter estimation:

  * (i)

the data $d$,

  * (ii)

the noise PSD $S_{n}(f)$,

  * (iii)

the waveform model $h(\theta)$,

as well as the main assumptions:

  * (iv)

the detector noise is stationary, leading to a diagonal noise covariance
matrix and an inner product that is a one-dimensional frequency integral, and

  * (v)

the detector noise is gaussian, which leads to the gaussian functional form of
the likelihood.

Each of the above introduces systematic uncertainties that will affect
inference at some level.

##  III Assumption: Gaussian noise

The functional form of the likelihood is dictated by the assumption of
gaussian detector noise. Gaussianity can be violated by instrumental
artifacts, known as glitches, or multiple GW signals temporally overlapping.
Glitches are a common occurrence, with a rate of $\lesssim 1$ per minute in
the LIGO detectors in O3a Abbott  _et al._ (2020c) and already coinciding with
signals, notably GW170817 Abbott  _et al._ (2017a); Pankow  _et al._ (2018).
Overlapping signals are expected to be rare in advanced LIGO but a possibility
for 3G detectors Pizzati  _et al._ (2021); Samajdar  _et al._ (2021); Relton
and Raymond (2021); Antonelli  _et al._ (2021).

The temporal coincidence of glitches and signals has led to the development of
mitigation techniques that simultaneously model the signal and the glitch
Chatziioannou  _et al._ (2021) or use auxiliary channel information Tiwari
_et al._ (2015); Meadors  _et al._ (2014); Ormiston  _et al._ (2020); Mogushi
(2021). In the context of tidal inference, glitches are relevant when
overlapping with the signal at frequencies $\gtrsim$400Hz. Though O3a was
dominated by glitches with peak frequencies below 100Hz Abbott  _et al._
(2020c), improved detector sensitivity could bring new glitch families. A
prominent glitch will be modeled together with the signal Chatziioannou  _et
al._ (2021), leading to unbiased signal parameters. However, this does not
preclude the possibility of a stealth bias Vallisneri and Yunes (2013), where
the glitch is not loud enough to be identified but could still affect tidal
inference.

Figure 2: Marginalized posterior for $\tilde{\Lambda}$ examining the effect of
nongaussian residuals (top) and misestimated noise PSD (bottom). Black
vertical lines give the injected values. Purple (orange) shade distributions
correspond to simulated signals of SNR 140 (420) in data with glitches. Green
shade distributions correspond to signals analyzed with different misestimated
high-frequency PSDs. Legends give the glitch SNR and the identification
probability, defined as the percentage of posterior samples that model the
glitch (top), and the level of PSD misestimation (bottom).

We explore this possibility by simulating BNS signals with SNR 140 and 420 in
a zero-noise realization. We break the gaussianity assumption by adding a
glitch in the LIGO-Livingston data given by a sine-gaussian that overlaps with
the BNS signal at 700Hz. The quality factor of the glitch is 20, resulting in
an instrumental transient that resembles the underlying signal but is present
in only one detector. We vary the SNR of the glitch and simultaneously model
the signal with waveform templates and the glitch with wavelets Chatziioannou
_et al._ (2021).

Figure 2 shows the resulting $\tilde{\Lambda}$ posterior also compared to the
case of no glitch and purely gaussian noise. Uncertainties are consistent with
the signal SNR and the projections of Fig. 1. In all cases the correct value
is recovered regardless of the SNR of the glitch. The skewed shape of the
posterior is due to a correlation between $\tilde{\Lambda}$ and the binary
mass ratio; we have verified that the two-dimensional posterior peaks at the
injected value. For glitches with $\rho\gtrsim$6 the probability of glitch
identification is $\sim$1\. For lower glitch SNR the probability drops, with
only $3\%$ of the posterior samples identifying a glitch of $\rho$=4. Despite
this, the $\tilde{\Lambda}$ posterior remains unbiased, suggesting no stealth
bias: if the glitch is too quiet to be identified, it is also too quiet to
bias tidal inference. This example considers one glitch and signal and it is
possible that different glitch morphologies could prove more problematic.
However, it shows that glitch mitigation techniques are already in place and
able to handle instrumental artifacts in the data.

The other possibility is that of a secondary signal overlapping with the
primary BNS. The most problematic scenario occurs for signals separated by
less than $0.1$s Pizzati  _et al._ (2021); Samajdar  _et al._ (2021); Relton
and Raymond (2021); Antonelli  _et al._ (2021) hence the secondary signal
could overlap with the tidally-affected region of the primary signal. Biases
could occur also if the secondary signal is subthreshold and not individually
resolvable Relton and Raymond (2021). This case requires a joint analysis of
the two signals, similar to the joint signal and glitch analysis from above,
and also similar to techniques developed for space-based detectors whose data
contain millions of overlapping signals Cornish and Crowder (2005); Littenberg
(2011); Littenberg  _et al._ (2020).

In summary, nongaussian features in the data are unavoidable, but they can be
modeled simultaneously with the primary signal to ensure unbiased parameter
recovery. What is more, any systematic bias from nongaussian data will be
unique to each BNS. The bias will therefore not accumulate when multiple
events are combined, though it might affect individual very loud sources.

##  IV Assumption & Ingredient: Stationary noise with a known PSD

The inner product of Eq. 2 is based on the assumption that the gaussian
detector noise is stationary: its mean and covariance are constant in time for
the duration of the analyzed data segment. The frequency-domain covariance
matrix is then diagonal with a (assumed known) variance related to $S_{n}(f)$.
Neither noise stationarity nor a perfect knowledge of $S_{n}(f)$ are strictly
true, introducing a potential systematic in parameter estimation.

Tidal effects influence the signal for a few tens of milliseconds, during
which the noise is most likely stationary. However, the entire signal lasts
for longer as a GW170817-like signal is 2(5){10} minutes from merger at
23(16){12}Hz, putting a strain on the stationarity assumption. Efforts to
subtract nonstationary noise Vajente  _et al._ (2020); Yu  _et al._ (2021) or
correct for it at the analysis level Zackay  _et al._ (2019) are under way,
with a further option of abandoning the frequency domain altogether Isi and
Farr (2021); Cornish (2020). Spectral lines in the data Littenberg and Cornish
(2015); Abbott  _et al._ (2020d); Huang  _et al._ (2020) and finite analysis
segments Talbot  _et al._ (2021) can also introduce non-diagonal terms in the
covariance matrix.

Under the assumption of stationarity the noise PSD itself can be computed
using either off-source Veitch  _et al._ (2015); Romero-Shaw  _et al._ (2020)
data that assume stationarity from even longer data segments or it can be
modeled based on onsource data Littenberg and Cornish (2015); Chatziioannou
_et al._ (2019). Uncertainty in the PSD estimation can also be marginalized
over Rover  _et al._ (2011); Rover (2011); Talbot and Thrane (2020);
Chatziioannou  _et al._ (2021). PSD errors could cause parameter biases
Chatziioannou  _et al._ (2019), however these should primarily affect the
amplitude of the GW signal and less so its phase evolution.

To test this we simulate a BNS in gaussian noise and analyze it with a
misestimated noise PSD in the $(400,1000)$Hz range. The PSD is based on the
LIGO design sensitivity and we alter its strain sensitivity in the relevant
frequency range by some percentage compared to the true value used for the
simulated data. The resulting $\tilde{\Lambda}$ posterior is shown in Fig. 2
showing PSD misestimation does not affect tidal parameter recovery for PSD
relative errors of up to $\pm 10\%$.

##  V Ingredient: Data

The data $d$ correspond to the relative displacement of the interferometer
test masses, tracked through interfering laser light incident on
photodetectors. Converting the photodetector output to strain is achieved
through a calibration process whose uncertainties could affect parameter
estimation if left unaccounted for Lindblom (2009); Vitale  _et al._ (2012).
Detector calibration relies on detector strain induced by photon calibrators
Karki  _et al._ (2016), resulting in an estimate for the systematic error and
corresponding statistical uncertainty for the detector frequency-domain
amplitude and phase response Abbott  _et al._ (2017c); Cahillane  _et al._
(2017); Sun  _et al._ (2020, 2021); Acernese  _et al._ (2021).

During first half of O3 the calibration uncertainty (systematic and
statistical) was determined to be no more than 4∘ in phase in the LIGO
detectors at the 68% level, corresponding to 7∘ at the 90% level Sun  _et al._
(2020), with similar estimates for the second half of O3 Sun  _et al._ (2021).
A conservative phase calibration error of 10∘ is compared in Fig. 3 against
the GW phase shift for different NS radii. The phase calibration uncertainty
is comparable to the GW dephasing induced by a $100-200$m change in the
radius.

Astrophysical parameter estimation studies marginalize over calibration
uncertainty Abbott  _et al._ (2016), a procedure that effectively increases
measurement uncertainties, though the result is small at current
sensitivities. Though calibration uncertainty is typically treated as being
uncorrelated between different frequencies, using a physical calibration model
that correctly encodes calibration error across frequencies could further
mitigate the effect on parameter estimation Vitale  _et al._ (2021); Payne
_et al._ (2020). Improvements in photon calibration Bhattacharjee  _et al._
(2021), alternative methods such as the Newtonian calibrator Estevez  _et al._
(2018); Ross  _et al._ (2021), and even astrophysical calibration Pitkin  _et
al._ (2016); Essick and Holz (2019) could reduce the impact of calibration
error which is currently comparable to the target radius uncertainty of
$100$m.

##  VI Ingredient: Waveform model

The final ingredient of the analysis, and the most commonly considered one in
the context of systematics, is the waveform model Dietrich  _et al._ (2021).
The effect of waveform systematics has been investigated for GW170817 by
employing a diverse set of waveform models including post-Newtonian Flanagan
and Hinderer (2008); Vines  _et al._ (2011), effective-one-body Damour and
Nagar (2009); Bernuzzi  _et al._ (2015); Nagar  _et al._ (2018); Gamba  _et
al._ (2020); Hinderer  _et al._ (2016); Steinhoff  _et al._ (2016), and
phenomenological models calibrated to numerical relativity simulations
Dietrich  _et al._ (2017, 2019, 2019); Kawaguchi  _et al._ (2018). The main
conclusion is that current statistical uncertainties dominate over waveform
systematics Abbott  _et al._ (2019a, 2018b).

Studies of simulated signals suggest, however, that waveform systematics could
become significant for $\rho\gtrsim$100 Dietrich  _et al._ (2021); Dudi  _et
al._ (2018); Samajdar and Dietrich (2019); Gamba  _et al._ (2021),
corresponding to GW170817 at design sensitivity or $50$ sources at the A+
timescale. Waveform biases increase with $\tilde{\Lambda}$, and thus less
massive or bigger NSs, and could be due to modeling error in the point-
particle or the tidal sectors of the waveform Dietrich  _et al._ (2021).
Additionally, numerical errors in the numerical relativity simulations used to
calibrate the waveform models could influence results at a similar SNR Dudi
_et al._ (2018) as in some cases waveform and simulation errors can be
comparable Bernuzzi  _et al._ (2012); Bernuzzi and Dietrich (2016); Kiuchi
_et al._ (2017); Dietrich  _et al._ (2018a); Dietrich and Hinderer (2017);
Dietrich  _et al._ (2018b). This level of systematic error will be comparable
to statistical errors for detectors with A+ sensitivity, once the statistical
uncertainty in $\tilde{\Lambda}$ ($R$) reaches 200 (0.5-1km) at the 90%
credible level.

We quantify the waveform accuracy required to achieve the radius measurement
projected for Voyager and 3G detectors in Fig. 3. Using the
IMRPhenomD_NRTidalv2 Dietrich  _et al._ (2019) waveform model we compute the
frequency-domain phase difference between signals emitted by BNSs of
$1.4M_{\odot}$ as a function of the NS radius relative to $13$km. A $0.5$km
radius difference induces a $3-4$rad dephasing; waveform systematics need to
be kept below this level to achieve such radius accuracy. Errors in numerical
simulations are typically quoted at $\sim$1rad Dietrich and Hinderer (2017),
though these refer to time-domain phase and are not directly comparable.

The dephasing is larger at higher frequencies where the detector sensitivity
decreases, thus making absolute phase differences less informative. As an
alternative we also plot a noise-weighted phase difference, derived in Sampson
_et al._ (2014) as the effective cycles of phase that a specific effect (here
the tidal deformation) contributes to the waveform. The effective cycles of
phase are related to an upper limit on the Bayes Factor that the effect in
question is detectable in the data Sampson  _et al._ (2014). We use the LIGO
design sensitivity, though the level of the PSD does not affect the noise-
weighted phase difference, only its shape.

Figure 3: GW frequency-domain phase difference between signals from BNSs with
different radii relative to $13$km up to different frequencies. Solid (dashed)
lines denote absolute (noise-weighted) phase differences. The shaded region
marks a $10^{\circ}$ calibration uncertainty, relevant only for the absolute
phase difference. Unbiased tidal and radius inference hinges on waveform
models achieving phase evolution accuracies better than the induced tidal
effect.

The noise-weighted phase difference is a less sensitive function of the upper
frequency cutoff, but it still increases for frequencies up to $1400$Hz. A
$0.5$km change in the radius leads to a noise-weighted phase change of
$\sim$0.3rad, again setting the threshold for waveform accuracy in other to
achieve such a radius measurement.

##  VII Conclusions

The astounding precision expected to be achieved with GW observations in the
next decade(s) Adhikari  _et al._ (2019) brings to the forefront all analysis
assumptions and ingredients to be examined under the light of systematic
errors. In the context of BNSs we study the frontier for measuring NS
properties by quantifying the limiting effect of various sources of
systematics. Improved waveform models are needed for a radius measurement of
$\Delta R\sim$1km at the timescale of A+, while improvements in detector
calibration are needed at the timescale of Voyager for $\Delta R\sim$200m.
Both waveform and calibration errors would introduce correlated biases for
different sources, and thus increasingly affect combined constraints.

The importance of nongaussian noise depends on the type of glitches in future
detectors, but methods to simultaneously model signals and glitches or
multiple signals will mitigate the effect. The effect of nonstationary noise
depends on the detector low frequency performance and hence the signal length.
Misestimating the noise PSD has a negligible effect on tidal inference.
Nongaussianity, nonstationarity, and PSD misestimation would likely affect
each signal in a unique way, and thus not accumulate in combined constraints.

This discussion concerns extracting binary parameters, namely masses and tidal
deformability, from GW data. The projected constraints were translated to NS
radius assuming a perfect conversion method for illustration. In reality,
additional analysis steps are required to combine information from multiple
sources and obtain constraints on NS radii or features such as phase
transitions Del Pozzo  _et al._ (2013); Agathos  _et al._ (2015);
Chatziioannou  _et al._ (2015); Han and Steiner (2019); Chatziioannou and Han
(2020); Han and Prakash (2020); Zhang and Li (2019); Pang  _et al._ (2020);
Drischler  _et al._ (2021). This is achieved through hierarchical inference
Mandel (2010); Mandel  _et al._ (2019) which accounts for parameters
correlations, statistical uncertainties, and selection effects. Additional
systematic errors here would be the NS equation of state model employed Legred
_et al._ (2021); Greif  _et al._ (2019), the estimation of the selection
function Abbott  _et al._ (2019b, 2021); Wysocki  _et al._ (2019); Farr
(2019), and the population model for NS spins and masses Chatziioannou and
Farr (2020); Landry and Read (2021). The latter, if neglected, could bias the
NS equation of state after 25-50 events Wysocki  _et al._ (2020) but the
effect can be fully addressed by simultaneously inferring the NS mass and spin
distribution with the equation of state Wysocki  _et al._ (2020); Golomb and
Talbot (2021).

Finally, this study examines only GW data and their analysis. X-ray
observations Miller  _et al._ (2019); Riley  _et al._ (2019); Miller  _et al._
(2021); Riley  _et al._ (2021); Bogdanov  _et al._ (2019a, b, 2021) will
provide additional information about NS properties yielding overall tighter
constraints. The relation between systematic errors in the different
observations and their effect on the decreasing statistical uncertainty needs
to be explored.

##  VIII Acknowledgements

We thank the participants of the Aspen Center program “Exploring Extreme
Matter in the Era of Multimessenger Astronomy: from the Cosmos to Quarks” for
numerous discussions that motivated this study. We thank Max Isi for comments
on the manuscript. This work was initiated at Aspen Center for Physics, which
is supported by National Science Foundation grant PHY-1607611. This work was
partially supported by a grant from the Simons Foundation. This research has
made use of data, software and/or web tools obtained from the Gravitational
Wave Open Science Center (https://www.gw-openscience.org), a service of LIGO
Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration.
Virgo is funded by the French Centre National de Recherche Scientifique
(CNRS), the Italian Istituto Nazionale della Fisica Nucleare (INFN) and the
Dutch Nikhef, with contributions by Polish and Hungarian institutes. This
material is based upon work supported by NSF’s LIGO Laboratory which is a
major facility fully funded by the National Science Foundation. The authors
are grateful for computational resources provided by the LIGO Laboratory and
supported by National Science Foundation Grants PHY-0757058 and PHY-0823459.
Software: gwpy Macleod  _et al._ (2020), matplotlib Hunter (2007), gwtoolbox
Yi  _et al._ (2021).

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