Inferring the population properties of binary neutron stars with gravitational-wave measurements of spin
Abstract
The recent LIGO-Virgo detection of gravitational waves from a binary neutron star inspiral event GW170817 and the discovery of its accompanying electromagnetic signals mark a new era for multi-messenger astronomy. In the coming years, advanced gravitational-wave detectors are likely to detect tens to hundreds of similar events. Neutron stars in binaries can possess significant spin, which is imprinted on the gravitational waveform via the effective spin parameter . We explore the astrophysical inferences made possible by current and future gravitational-wave measurements of . First—using a fiducial model informed by radio observations—we estimate that of binary neutron stars should have spins measurable by advanced detectors () assuming the spin axis of the recycled neutron star aligns with the total orbital angular momentum of the binary. Second, using Bayesian inference, we show that it is possible to distinguish between our fiducial model and some extreme initial spin period distributions using detections. Third, with detections, we can confidently tell whether or not the spin axis of the recycled neutron star tends to be aligned with the binary orbit. Finally, stringent constraints can be placed on neutron star magnetic field decay after detections, if the spin periods and magnetic field strengths of Galactic binary neutron stars are representative of the merging population.
I Introduction
On August 17, the LIGO Aasi et al. (2015) and Virgo Acernese et al. (2015) detectors observed GW170817, the first gravitational-wave (GW) signal from a binary neutron star (BNS) merger Abbott et al. (2017a). The discoveries of an electromagnetic counterpart, visible across the electromagnetic spectrum, ushered in the highly-anticipated era of multi-messenger astronomy Abbott et al. (2017b); Andreoni et al. (2017). Improved merger rate estimates now indicate that around 10–200 detections of BNS mergers are expected per year of operation at design sensitivities for Advanced LIGO and Virgo Abadie et al. (2010); Abbott et al. (2017a). This will enable detailed studies of the population properties of BNS systems.
In 1974, the first pulsar in a BNS system PSR B1913+16 was discovered in radio pulsar surveys Hulse and Taylor (1974, 1975). There are now 15 BNS systems known in our Galaxy Tauris et al. (2017). Of particular significance is the Double Pulsar PSR J07373039A/B Burgay et al. (2003); Lyne et al. (2004). It remains the only BNS system in which both stars can be observed as radio pulsars. In the standard scenario, the recycled NS in a BNS system gets spun up by accreting matter and angular momentum from its companion Tutukov and Yungelson (1973); Flannery and van den Heuvel (1975); Smarr and Blandford (1976); Kalogera et al. (2007); Postnov and Yungelson (2014). The final spin period depends on the rate and duration of the accretion process. There is a tendency for the spin periods of the recycled pulsars to be shorter in closer systems Tauris et al. (2017). This can be attributed to the fact that pulsars in wider binaries generally have less time to gain angular momentum before their companion stars detonate as supernovae.
It is generally believed that NSs spin down due to the loss of rotational energy by powering magnetically driven plasma winds Goldreich and Julian (1969); Contopoulos et al. (1999); Spitkovsky (2006). Magnetic fields play a key role determining the astrophysical character of a NS (see Bransgrove et al. (2017) and references therein). Understanding the evolution of magnetic fields will give insights into both the physics of NS interiors and the observed population of pulsars in the Galaxy. In this paper, we demonstrate that GW measurements of NS spins may shed new light on NS magnetic field evolution.
The spin of merging compact objects is imprinted on the gravitational waveform. While there are six degrees of freedom describing the two spin vectors, the gravitational waveform depends primarily on a single combination of parameters called the effective spin parameter Ajith et al. (2011); Cutler et al. (1993)
(1) |
Here, are the angles between the spin vectors and the orbital angular momentum vector, is the mass ratio of component masses , and are the dimensionless spin magnitudes of the merging objects Abbott et al. (2016); Dietrich et al. (2015); Brown et al. (2012),
(2) |
Here , is the NS moment of inertia, and is the angular spin frequency. Physically, is the mass weighted sum of the spin projections along the axis of the orbital angular momentum. As is increased above zero, the merger takes place at a higher frequency, and produces a relatively longer signal in the observing band. Conversely, as is made increasingly negative, the merger takes place at a lower frequency, producing a relatively shorter signal.
In the case of stellar-mass binary black hole mergers, it has been shown that GW measurements of spin can be used to distinguish different formation channels (see, e.g., Vitale et al. (2017); Stevenson et al. (2017); Talbot and Thrane (2017); Farr et al. (2017)). Black holes formed through isolated binary evolution are expected to spin in alignment with the binary orbital angular momentum, whereas the spin orientation of black holes in dynamically formed binaries is likely to be isotropically distributed. By combining multiple detections, it is possible to constrain the typical misalignment angle of black hole spins and infer the relative fraction of different sub-populations.
While published GW measurements of GW170817 are consistent with Abbott et al. (2017a), advanced detectors operating at design sensitivity may be able to measure statistically significant spin in BNS inspirals. In this paper, we point out that a significant fraction of future LIGO-Virgo detections may contain measurable signatures of spin. Combining multiple measurements from an ensemble of BNS signals, it will be possible to constrain the shape of the distribution from which we can infer interesting NS population properties. In particular, we show that GW measurements can be used to constrain (1) the typical misalignment angle between the NS spin axis and the orbital angular momentum and (2) the decay timescale of NS magnetic fields.
The organization of this paper is as follows. In Section II, we present a fiducial model of NS spin distribution that we use to investigate how GWs might inform our understanding of NS evolution. The model is built to be consistent with radio observations, but it does not necessarily cover the full range of possible scenarios. Additional details are provided in Appendices A and B. In Section III, we discuss the sensitivity of advanced detectors to measure NS spins. In Section IV, we present a Bayesian framework for inference of BNS population properties. We demonstrate model selection and hyper-parameter estimation using Monte-Carlo data. Finally, in Section V, we provide concluding remarks.
Ii Fiducial Models
In order to link GW measurements to NS evolution, we need to know : the probability density function for the effective spin parameter at merger given a model with population hyper-parameters . The idea is that by measuring , we can make inferences about the NS evolutionary properties described by the hyper-parameters . For example, we investigate our ability to constrain the NS magnetic field decay timescale . Here we present a suite of fiducial models that represent plausible distributions of for illustrative purposes.
We construct four fiducial models, labeled as Standard, Iso Spin, B Decay, and Diff EOS. Each model corresponds to a different vector of hyper-parameters . We provide an overview of the fiducial models for . More details on the construction of these models are given in Appendices A and B.
To construct each fiducial model, we first construct the probability density function for a vector of parameters describing the BNS system at the time of its birth.
These parameters include: component masses, spin periods, magnetic field strengths, binary orbital period and eccentricity.
This eight-dimensional distribution is constructed using observations of known Galactic BNS systems.
Next, we evolve the at-birth parameters forward in time in order to obtain a new distribution , which describes these same parameters at the time of merger.
In order to carry out this calculation, we assume that NSs spin down due to magnetic dipole braking
It is useful to note that the vector of hyper-parameters describes the population of BNSs. This may be contrasted with parameters , which describe the properties of an individual binary (at birth and at merger respectively). The equation of state is an example of a hyper-parameter since it is a property of the population as a whole. The NS masses are an example of a parameter since they are different for each BNS.
Our standard model assumes that the spin axis of the recycled NS is aligned with the total binary orbital angular momentum. That is, . This is a plausible assumption given that the only measured misalignment angle (which comes from the Double Pulsar) is small: at 95% confidence, assuming that the observed emission comes from both magnetic poles which is the scenario favored by radio data Ferdman et al. (2013). We assume that the magnetic field decay time is long compared to the age of the Universe. We assume the AP4 equation of state Lattimer and Prakash (2001).
The Iso Spin model is identical to the Standard model except we assume that is drawn from a uniform distribution. While this model is disfavored by the Double Pulsar, we include it in order to demonstrate how GW measurements can help distinguish between Standard and Iso Spin.
The B Decay model is identical to Standard, except that we assume an exponentially decaying magnetic field
The Diff EOS model is identical to Standard, except that we use the PAL1 equation of state instead of AP4 (see Lattimer and Prakash (2001) and references therein). For a mass of , the NS radius is for PAL1 compared to for AP4. Therefore, larger values of are predicted for PAL1 compared to the AP4 equation of state all else equal.
Model | B-field decay | EOS | note | |
---|---|---|---|---|
Standard | aligned | no | AP4 | Empirical |
Iso Spin | isotropic | no | AP4 | Empirical |
B Decay | aligned | AP4 | Empirical | |
Diff EOS | aligned | no | PAL1 | Empirical |
PopSynth1 | aligned | yes | AP4 | Pop. Synth. |
Pulsar in | Chirp mass | |||
---|---|---|---|---|
BNS systems | () | () | () | () |
J17571854 | 10.0 | 14.4 | 4.84 | 1.189 |
J07373039A/B | 9.5 | 14.0 | 7.39 | 1.125 |
J1913+1102 | 8.3 | 12.1 | 0.32 | 1.243 |
B2127+11C | 4.5 | 8.3 | 3.19 | 1.181 |
J17562251 | 4.2 | 11.6 | 0.48 | 1.118 |
B1913+16 | 2.1 | 4.3 | 2.44 | 1.231 |
B1543+12 | 1.9 | 7.9 | 0.34 | 1.166 |
In addition to the four fiducial models, we also consider a population synthesis model from Osłowski et al. (2011) denoted here as PopSynth1. In this model, if a common envelope phase occurs before the formation of the second NS, there is an episode of hyper-critical accretion. This allows a large amount of matter to be accreted compared to other models in Osłowski et al. (2011). In this model, the magnetic field decays until reaching a minimum of . As a consequence, roughly half of the recycled NSs are spun up during a subsequent Roche-lobe overflow phase to spin periods of , while the rest spin at a much slower rate; see the model hp in Fig. 3 in Ref. Osłowski et al. (2011).
The five models are summarized in Table 1. The corresponding distributions are shown in Fig. 1. Each colored curve corresponds to a different model. For reference, we also plot as a shaded vertical band the range of expected at merger for the 7 Galactic BNS systems that consist of a recycled pulsar and that will merge within a Hubble time. We evolve the spin periods forward until the merger, which will happen in between and . We consider two cases. We calculate assuming no magnetic field decay and using the AP4 equation of state. We calculate assuming the PAL1 equation of state and magnetic field decay with a timescale of . The results are listed in Table 2. Also included are the chirp mass and the relative merger rate defined as the reciprocal of the sum of pulsar characteristic age and binary coalescence time. We use the B pulsar’s age when calculating for the Double Pulsar. Details of the calculation and information about all known Galactic BNS systems are given in Appendices A and B. We assume the other systems are similar to the Double Pulsar, i.e., the spin axis of the recycled pulsar is aligned with the binary orbit and the companion NS has effectively zero spin. Out of 7 binaries, at least 2 are expected to have .
There are a number of interesting features worthy of remark in Fig. 1. First, for all five models, a significant fraction of BNS (-) are characterized by relatively large spins with . We note that, for our four fiducial models, essentially all of the events possess , consistent with the “low-spin” prior used in the GW170817 discovery paper Abbott et al. (2017a).
Second, we see that PopSynth1 predicts a dramatically different distribution. While half of the population has , the other half possess much larger spins with . We note that such high spins are not supported by current radio observations for pulsars with NS or massive white dwarf companions. We show that it is possible to definitively test PopSynth1 with a small number of detections in section IV.1.
Third, there are clear differences between different fiducial models. In the Standard model, is found to be between 0 and 0.03 with probability. In the Iso Spin model, the distribution is symmetric with respect to zero spin and . B Decay and Diff EOS predict higher values of than Standard, extending the high end to 0.04. For Diff EOS, this is because of larger NS radii allowed by PAL1 than the AP4 equation of state. For B Decay, this is because of lower spin-down rates. The peak position of B Decay depends on the magnetic field decay timescale, which can be constrained using a large number of BNS detections given our prior knowledge of the equation of state. We demonstrate this in Section IV.2.
Iii Measurement uncertainty of spin using advanced detectors
In order to compare the distributions of to the sensitivity of advanced detectors, we carry out Monte-Carlo simulations.
We inject a series of BNS signals into Gaussian noise generated assuming Advanced LIGO design sensitivity
We simulate nine events equally spaced between and . The masses are set to , matching the Double Pulsar. The right ascension (), declination (), and GPS time (1187008882) are chosen to match GW170817. The effective spin parameter is set to , which is at the high end of the distribution in our fiducial models. The inclination angle , the polarization angle , and the phase at coalescence are all set to zero. Our choice of source parameters is somewhat arbitrary, but the purpose of the LALInference runs is to obtain the approximate scaling law between the realistic measurement uncertainty of and the signal-to-noise ratio (). The exact choice of source parameters does not significantly impact this scaling law.
An example posterior distribution of produced using LALInference is provided in Fig. 2. This event is simulated at a distance of 200 Mpc. The expected for a LIGO Hanford-Livingston detector network is 17.4. The posterior samples are well fit by a Gaussian distribution with mean 0.021 and standard deviation, (blue curve). In general, we find that the marginalized posterior distribution of from LALInference runs on simulated data can be well approximated by a Gaussian distribution.
In Fig. 3, we plot as a function of , which is well approximated by the scaling law Cutler and Flanagan (1994). Using this relation, we can speed up subsequent simulations by approximating the reduced likelihood function as a Gaussian distribution with width
(3) |
where the mean is the maximum-likelihood estimate for a particular random noise realization. Each value of is generated by adding a normally-distributed random number (variance = ) to the true value of .
We simulate BNS sources that are uniformly distributed in volume (i.e., uniform in distance cubed) out to 300 Mpc. Note that within such a distance we can safely ignore the merger rate evolution over redshift. The other source parameters such as sky location and inclination angle are appropriately randomized according to suitable distributions.
We calculate the (expectation value of) for the Advanced LIGO Hanford-Livingston detector pair. Events with a network matched-filter 12 are excluded because they fall below the threshold necessary for unambiguous detection Abadie et al. (2012). This leads to about 2000 detected events. As expected, the distribution of is consistent with Schutz (2011). The median in our Monte Carlo dataset is 15, which corresponds to an effective spin measurement uncertainty of (see Fig. 3).
For comparison, we also show the GW170817 uncertainty for the low-spin prior as a black square in Fig. 3.
The numerical value is computed by approximating the published 90% confidence () interval Abbott et al. (2017a) with a Gaussian distribution.
At the same , the measurement sensitivity of given by our scaling law (depicted by the blue curve in Fig. 3) is a factor of 2 better than that of GW170817. This may be due to the combined effects of 1) better sensitivity at low frequency, which is critical to measure the GW phase evolution and thus spin effects, 2) systematic errors of waveform models and 3) calibration uncertainties of real data.
Specifically, for the analysis of GW170817 Abbott et al. (2017a), i) a different waveform model was used to account for both tidal effects and spins, which leads to different statistical uncertainties for and ii) the subtraction of the glitch that was superposed on the signal in the LIGO-Livingston data could have impacted the uncertainties for parameter estimates. Furthermore, our scaling law for is derived using simulations with a single mass ratio (), whereas for GW170817 there is significant support for smaller values of . Since of our simulated BNSs have
We compare Advanced LIGO’s sensitivity to measure with the effective spins of Galactic BNS systems at the time of merger (Table 2). In Fig. 4, we show the luminosity distance out to which can be measurable at the 2- level. Blue open circles are Galactic BNS systems, taking the lower values of listed in Table 2. We assume an optimal binary orientation () and all systems are placed at the sky location of GW170817. The size of the blue circles are proportional to the relative merger rate (see Table 2) of the particular system. Red squares with downwards arrows indicate the upper limit for GW170817 and the expected limit at design sensitivity. Note that the GW170817 design limit lies above the linear trend of blue circles; the distance is about 0.005, which can be attributed to the asymmetric 90% confidence interval Abbott et al. (2017a).
To put Fig. 4 into an astrophysical context, we add a second y-axis for the spin frequency of the recycled NS in Fig. 4. The conversion from to the spin frequency employs the average mass of 7 recycled pulsars and , i.e., assuming and ; see Eqs. (1-2). We show that Advanced LIGO operating at design sensitivity will be sensitive to up to Mpc. Out of 7 known Galactic BNS systems, 2 () are above this threshold assuming aligned spin. Given the relatively higher merger rate for these two systems, an even larger fraction of merging BNS should have measurable spins.
Iv Bayesian inference
In this section, we present a Bayesian formalism to infer BNS population properties from GW measurements of . In subsection IV.1, we demonstrate how GW measurements from an ensemble of detections can be used to carry out model selection, thereby allowing us to differentiate between the different models in Table 1. In subsection IV.2, we show how to constrain the magnetic field decay timescale using a large number of detections.
iv.1 Model selection
Let be the likelihood of GW data given BNS parameters . The GW parameter space includes 15 or more parameters including, e.g., inclination angle, luminosity distance, and component masses. Here, however, we are only concerned with . Thus, we marginalize over every parameter except to obtain a marginalized likelihood function .
The Bayesian evidence for a model given the data is
(4) |
where is the conditional prior distribution for given a model {Standard, Iso Spin, Diff EOS, PopSynth1, B Decay}. Rewriting Eq. 4 in terms of posterior samples MacKay (2002), we obtain
(5) |
Here, the index runs over posterior samples, each corresponding to a different value of ; see also Eq. (7) in Talbot and Thrane (2017). is the prior used by LALInference in order to obtain the initial set of posterior samples. It appears in Eq. 5 in order to convert the posterior distribution output by LALInference into a likelihood. Since Model PopSynth1 predicts a significant fraction of BNS events with , is set as with a flat distribution for both and in throughout this work. This is wider than the low-spin prior used for actual LALInference simulations. This makes our results slightly optimistic by less than a factor of 2. We simulate mock posterior samples using the Gaussian likelihood function defined in Eq. (3).
The total evidence from detections is simply the product of the evidence for each detection:
(6) |
The Bayes factor (BF) comparing model and model is given by
(7) |
Following convention MacKay (2002), we impose a threshold of () to define the point beyond which one model is significantly favored over another.
In order to demonstrate model selection, we carry out a Monte Carlo simulation as described in Section III. We use the scaling relation between and shown in Fig. 3 and the Gaussian likelihood function in Eq. 3. The calculation assumes a two-detector Advanced LIGO network operating at design sensitivity.
Then, we calculate three different Bayes factors (plotted in Fig. 5 as a function of the number of events): Standard / Iso Spin (black), Standard / PopSynth1 (red) and Standard / Diff EOS (blue). The colored lines show the median value while the shaded region indicates the 1- confidence interval.
When , we are able to clearly distinguish between the Standard model and the three alternatives considered here. The red curve shows that advanced detectors can distinguish between Standard and the (extreme) distribution of PopSynth1 after only detections. The black curve shows that advanced detectors can distinguish between Standard and Iso Spin after events. Finally, the blue curve shows that advanced detectors can distinguish between Standard and Diff EOS after events.
iv.2 Constraining magnetic field decay
Given a NS equation of state, one can construct a posterior for magnetic field decay timescale
(8) |
Here is the prior distribution for , which is assumed to be log-uniform on the interval . Note that such a wide prior is used only as an example to test our method and does not fold in any theoretical or observational information. Our primary purpose here is to demonstrate how many GW detections are required to place stringent constraints on using GW data alone. The distribution is the conditional prior for given model and the decay timescale . An example conditional prior is represented visually in Fig. 1 for Model B Decay with Myr.
Marginalizing over , we obtain a posterior on . In practice, we do not carry out an integral over a continuous variable; we sum up posterior samples. The discrete version for Eq. (8) is
(9) |
Generalizing to events, the posterior for becomes
(10) |
Here, represents the posterior sample from the event.
Following the same procedure described in Subsection IV.1, we generate an ensemble of detections. Events are generated using the B Decay model with the true value of Myr. In order to infer using Eq. (10), we generate for a vector of equally spaced in log space between 1 Myr and 10 Gyr.
Fig. 6 shows the posterior distribution inferred from 200 events. The true magnetic field decay timescale is marked with a red vertical line. We repeat the simulation for 100 times to account for random noise fluctuations. The typical 1- variations are shown as the shaded region in Fig. 6. For of noise realizations, the true value of is recovered within the 2- confidence interval. In this example, we find that can be constrained to be between and with confidence with 200 detections.
V Discussion and Conclusions
The detection of the BNS inspiral event GW170817 has opened up new opportunities to study NS physics. In the next five years, observations of similar events will become increasingly more frequent as Advanced LIGO/Virgo approach their design sensitivities and new detectors such as KAGRA Somiya (2012) and LIGO-India Unnikrishnan (2013) start to operate. As advanced detectors observe tens, and eventually hundreds, of BNS inspirals, it will be possible to make inferences about NS population properties using GW observations. We demonstrate that GW measurements of spin will have implications for the minimum NS spin period for BNSs after events, for the typical spin tilt angle of NSs after tens of detections, and for the NS magnetic field evolution after hundreds of observations.
In this proof-of-concept analysis, we rely on a set of fiducial models. The models are constructed to be consistent with radio pulsar observations, but they do not necessarily represent the full range of allowable parameter space. While the fiducial models are suitable for illustrative purposes, the next step in this program is to develop a more nuanced model, which does allow for the full range of possible initial conditions. It should be possible to then combine all available radio and GW data within a single Bayesian framework, yielding optimal constraints on NS evolution.
Our fiducial model does not account for selection effects associated with detecting radio pulsars in BNS systems. We assume that Galactic BNSs detected in radio are representative of merging binaries detectable with GWs. That is, pulsar luminosities are independent of their spin periods and magnetic field strengths. In practice, even though such an assumption holds, pulsars with shorter spin periods are more difficult to find in radio surveys. In future studies, it will be necessary to carefully include all selection effects in a formulation that attempts to combine GW and radio data.
We find that 2 out of 7 () of the known BNS systems that will merge within a Hubble time have spins that are marginally detectable up to 140 Mpc by Advanced LIGO operating at design sensitivity. The true fraction of BNSs with measurable spins may be higher since short-lived binaries like those two systems are less likely to be observed in radio. Our standard fiducial model assumes the spin axis is aligned with the binary orbital angular momentum and we do not consider spin precession effects. While this is a second-order effect, more information can be extracted by searching for precessing binaries.
In this work, we focus on BNS systems formed via the isolated binary evolution channel. It is generally believed that there is insufficient time for the recycled NS to reach ms spin periods and the second NS spins down much faster and thus makes no contribution to the effective spin at the time of binary merger. While our fiducial models predict that essentially all merging BNS have , binaries formed via dynamical captures could have much larger spins. Measurements of NS spins outside our fiducial model prediction will have interesting implications about their formation history.
Finally, our analysis does not account for NS tidal effects. Although there is not strong covariance between and NS tidal parameters, simultaneously fitting for both spins and tides would result in different statistical errors for measurements. Comparing our assumed sensitivity with spin constraints for GW170817 indicates that our results might be overly optimistic, but by no more than a factor of two. Additionally, we find there is a degeneracy between NS equation of state and magnetic field evolution for the distribution of BNS mergers. Constraints on NS magnetic field decay using the analysis method proposed in this paper would rely on prior knowledge on the equation of state. After tens to hundreds of BNS detections, measurements of NS tidal effects and possibly post-merger signatures will place tight constraints on the equation of state and thus allow us to study NS magnetic field evolution with spin Lackey and Wade (2015).
Acknowledgements.
We thank Matthews Bailes, Salvatore Vitale, Simon Stevenson and Christopher Berry for helpful comments on the manuscript and Colm Talbot, Rory Smith, Alexander Heger, Bernhard Müller, Sylvia Biscoveanu, Lijing Shao, Shuxu Yi and Gang Wang for useful discussions. XZ, ET & YL are supported by ARC CE170100004. ET is supported through ARC FT150100281. SO acknowledges support through the ARC Laureate Fellowship grant FL150100148. PDL is supported through ARC FT160100112 and ARC DP180103155. This is LIGO Document LIGO-P1700400.Appendix A NS spin evolution and BNS inspirals
Here we describe the calculations necessary to evolve the at-birth parameters of BNS systems to the parameters at merger as outlined in Section II. We construct the distribution of these parameters, such as NS mass, spin and binary orbital period and eccentricity, using radio pulsar observations in section B.
The coalescence time of a circular binary due to the emission of GWs is Thorne (1987)
(11) |
where is the initial binary orbital frequency, is the chirp mass defined as , with being the total mass, and is the symmetric mass ratio. The coalescence time for an eccentric binary can be computed as Peters and Mathews (1963); Peters (1964)
(12) |
Here is the initial binary orbital eccentricity, the constant and the function are
(13) |
(14) |
For an eccentric binary, the orbital frequency and eccentricity co-evolve as
(15) |
where the function is defined as
(16) |
Evolving the BNS systems listed in Table 3 that will merge within a Hubble time forward in time, we find that they are expected to be in nearly circular orbits () when they enter the sensitive frequency band ( Hz) of ground-based interferometers in between 76 Myr and 2.7 Gyr.
Assuming magnetic dipole braking, the NS spin frequency evolution follows Ostriker and Gunn (1969):
(17) |
where with being spin period, . The torque parameter is defined as Spitkovsky (2006)
(18) |
where is the surface magnetic field strength, is the NS radius, is the moment of inertial, is the misalignment angle between the magnetic dipole moment and spin axis. There are a few notes worthy of mentioning here. First, Eq. 18 is a convenient approximation to numerical results found in Spitkovsky (2006). Second, there is a degeneracy between and . Since pulsar measurements collected here only allow us to infer , we choose to fix at . The exact value of is not important as the overall spin-down behavior is determined by . Third, the spin-down estimator of can be derived from where and is the spin period and period derivative respectively.
The moment of inertial can be computed as
(19) |
The above equation is found as an empirical relation by fitting to a sample of the equation of state Lattimer and Schutz (2005). In this work we consider two equations of state: AP4 and PAL1. While these two models do not cover the entire range of possible equations of state that survive astrophysical constraints (see Ref. Lattimer (2012) for a recent review), they are chosen for illustrative purposes. For a typical NS mass of , the NS radius is 11 km and 14 km for AP4 and PAL1 respectively.
Assuming remains a constant throughout the binary’s lifetime, it is straightforward to show that the spin angular frequency at coalescence is:
(20) |
where and are the initial values of angular spin frequency and its time derivative respectively.
It is possible that the NS magnetic filed decays exponentially over time Romani (1990); Geppert and Urpin (1994); Konar and Bhattacharya (1997); Cumming et al. (2004); Bransgrove et al. (2017)
(21) |
where is the magnetic field decay timescale. Under this decaying-B field model, the final spin frequency is
(22) |
Pulsar Name | (ms) | (ms) | (days) | (Gyr) | Age (Gyr) | B ( G) | () | () | Ref. | |||
systems will merge within a Hubble time | ||||||||||||
J17571854 | 21.50 | 24.29 | 0.023 | 0.020 | 0.184 | 0.606 | 0.076 | 0.13 | 1.3 | 1.338 | 1.395 | Cameron et al. (2017) |
J07373039A | 22.70 | 27.17 | 0.022 | 0.018 | 0.102 | 0.088 | 0.086 | 0.204 | 1.1 | 1.338 | 1.249 | Kramer et al. (2006) |
J07373039B | 2773 | 4596 | 0 | 0 | 0.102 | 0.088 | 0.086 | 0.049 | 260 | 1.249 | 1.338 | Kramer et al. (2006) |
J1913+1102 |
27.29 | 29.59 | 0.016 | 0.015 | 0.206 | 0.090 | 0.473 | 2.685 | 0.4 | 1.61 | 1.27 | Lazarus et al. (2016) |
J17562251 | 28.46 | 61.95 | 0.017 | 0.008 | 0.320 | 0.181 | 1.656 | 0.043 | 0.9 | 1.341 | 1.230 | Ferdman et al. (2014) |
B2127+11C | 30.53 | 54.90 | 0.016 | 0.009 | 0.335 | 0.681 | 0.217 | 0.097 | 2.2 | 1.358 | 1.354 | Jacoby et al. (2006) |
B1534+12 | 37.90 | 131.5 | 0.013 | 0.004 | 0.421 | 0.274 | 2.734 | 0.248 | 1.7 | 1.333 | 1.346 | Fonseca et al. (2014) |
B1913+16 | 59.03 | 114.6 | 0.008 | 0.004 | 0.323 | 0.617 | 0.301 | 0.109 | 4.1 | 1.440 | 1.389 | Weisberg et al. (2010) |
J1906+0746 | 144.1 | 7636 | 0.004 | 0 | 0.166 | 0.085 | 0.308 | 0.0001 | 289 | 1.291 | 1.322 | van Leeuwen et al. (2015) |
systems will not merge within a Hubble time | ||||||||||||
J18072500B | 4.19 | 150.8 | 0.117 | 0.003 | 9.957 | 0.747 | 1044 | 0.81 | 0.1 | 1.366 | 1.206 | Lynch et al. (2012) |
J1518+4904 | 40.93 | 792.4 | 0.013 | 0 | 8.634 | 0.249 | 8914 | 2.39 | 0.2 | Janssen et al. (2008) | ||
J1829+2456 | 41.01 | 95.96 | 0.013 | 0.005 | 1.176 | 0.139 | 55.42 | 1.24 | 0.2 | Champion et al. (2005) | ||
J0453+1559 | 45.78 | 885.1 | 0.010 | 0 | 4.072 | 0.113 | 1453 | 3.90 | 0.6 | 1.559 | 1.174 | Deneva et al. (2013) |
J17532240 | 95.14 | - | 0.005 | 0 | 13.64 | 0.304 | 1.55 | 1.7 | - | - | Keith et al. (2009) | |
J18111736 | 104.2 | - | 0.004 | 0 | 18.78 | 0.828 | 1862 | 1.83 | 1.9 | Corongiu et al. (2007) | ||
J19301852 | 185.5 | - | 0.003 | 0 | 45.06 | 0.399 | 0.16 | 9.7 | Swiggum et al. (2015) |
Parameter | min | max | distribution | note/ |
---|---|---|---|---|
correlation | ||||
3.8 | 49 | Uniform | Eq. (23) | |
0.01 | 0.5 | Uniform | ||
0.004 | 0.78 | Uniform | ||
8.0 | 10 | AP4 | ||
8.0 | 10 | PAL1 | ||
10 | 900 | Faucher-Giguère and Kaspi (2006); Ridley and Lorimer (2010) | ||
10.5 | 15 | Faucher-Giguère and Kaspi (2006); Ridley and Lorimer (2010) | ||
1.1 | 2.0 | NS radius | ||
1.0 | 1.6 | |||
Appendix B Construction of fiducial models
In this section we describe the construction of our models for the distribution of BNS parameters at birth . The models are fully motivated by pulsar observations of currently known Galactic BNS systems. These observations are summarized in Table 3, including current and final (i.e., when binary merges) spin periods, the corresponding dimensionless spin magnitudes, binary orbital period and eccentricity, coalescence time, characteristic age and surface magnetic field strength, masses of the pulsar and its companion.
In our fiducial model, each binary consists of a recycled NS and a “normal” NS. Here “normal” means that it is born with a period of but quickly spins down to seconds period over Lorimer and Kramer (2005). PSR J1906+0746 is considered to be normal (similar to an isolated pulsar) because it is a young ( kyr) pulsar with much higher spin-down rate Lorimer et al. (2006). As such, we consider its companion as a recycled NS. Table 4 summarizes information about our fiducial BNS population models. We describe the details below.
We first derive a model for NS mass distribution from 10 BNS systems that have precise component mass measurements. To represent the full range of mass ratio between the recycled NS and its companion, we fit separately a two-Gaussian model to 10 measurements. For the recycled NS, the main peak of the mass distribution is centered around 1.35 and the second around 1.57 , with weight of approximately and respectively. While the major Gaussian component is consistent with previous estimates (e.g., Özel et al. (2012); Kiziltan et al. (2013)), the minor peak is intuitively contributed by the two high mass systems PSR J0453+1559 and PSR J1913+1102. For the non-recycled NS, the major Gaussian component is centered around 1.24 and the second around 1.37 , with weight of about and respectively.
We assume is uniformly distributed in the range of days. This corresponds to the range of measured orbital periods of Galactic BNS systems that will merge within a Hubble time, except that we extend to day. Current searches for radio pulsars in relativistic binaries are limited to orbital periods longer than a few hours. The exact value of the lower cut for has no significant impact as the population is dominated by systems in wider orbits. We assume that the initial spin period () of the recycled NS and the orbital eccentricity () are also uniformly distributed but are correlated with as described below.
Figs. 7 and 8 provide an intuitive picture about the initial distributions of BNS parameters in our fiducial model. Blue dots mark the positions of synthetic BNS systems, whereas open stars represent known Galactic systems. Red (green) stars are for binaries that will (will not) merge within a Hubble time. Yellow open circles are obtained by evolving red stars back in time, indicating the initial conditions when the second NS was born.
In the lower panel of Fig. 7, open stars indicate positions of recycled pulsars in BNS systems in the orbital period – spin period diagram. As one can see, there exists a linear correlation for -
(23) |
Here we consider (the slope for black and red straight lines) as found by a fit to 11 Galactic BNS systems studied in Tauris et al. (2017). Such a linear correlation is further confirmed by the 13 BNS systems considered in this work. and are reference values used to determine the boundary region in the parameter space.
In order to obtain the initial values of , , and for 7 Galactic BNS systems at the birth of the second NS, we evolve these systems back in time assuming magnetic dipole braking and GW-driven orbital decay. The time is set as half of the characteristic age. Although this is somewhat arbitrary, it should be noted that the spin period approaches zero when evolving the pulsar back by the full amount of its characteristic age. The evolution is shown as yellow tracks in the bottom panel of Fig. 7 and Fig. 8. The upper bound (black line) in the - plane is set by PSR J17562251 with initial orbital period of 0.37 day and spin period of 43 ms. The lower bound (red line) is set by PSR J17571854 with initial orbital period of 0.32 day and spin period of 15 ms. The black and red lines, together with the cut-off in orbital period, determine the maximum and minimum values for to be 49 and 4 ms respectively.
In the top panel of Fig. 7, we show the restricted region in the - plane for our fiducial BNS population model. In the GW-driven regime, and co-evolve following equations (15-16). This is indicated by red and black curves, which cross J1913+1102 and J17571854 respectively. Such a choice excludes binary systems with either very short birth orbital periods and high eccentricities (top-left corner) or long periods and low eccentricities (bottom-right corner), as both are shown to be unlikely based on pulsar observations. Red and black curves, together with the cut-off in , lead to a range of between 0.004 and 0.8.
We make use of NS mass-radius relation described by the AP4 and PAL1 equation of state. The magnetic field strength of the recycled NS at the time of the birth of its companion is assumed to follow a log-Normal distribution with a mean value and a standard deviation for the AP4 (PAL1) equation of state. Such a distribution is found to be able to cover the scatter in the - diagram for recycled pulsars at the birth of second NS considered in this work, as demonstrated in Fig. 8. Because PAL1 allows relatively large NS radii, the required magnetic field strength to produce the same spin-down rate is lower; see Eq. (18). This illustrates the covariance between the equation of state and the magnetic field in probing NS spin evolution. Future GW measurements of NS tidal effects and possibly post-merger signatures are likely to place tight constraints on the equation of state. This will enable better understanding on NS magnetic field evolution through spin measurements.
For the second-born NS, we assume that i) the initial spin period follows a Gaussian distribution with a mean of and a standard deviation of Faucher-Giguère and Kaspi (2006); Ridley and Lorimer (2010); and ii) the initial magnetic field strength follows a log-Normal distribution with a mean and a standard deviation Faucher-Giguère and Kaspi (2006). Note that our results are insensitive to details of these two distributions as in our fiducial model the second NS makes no contribution to unless the magnetic field decay timescale is much shorter than binary coalescence time.
Finally, we briefly comment on directions to improve the fiducial models presented here. Our primary goal in developing these fiducial models was to cover parameter ranges informed by pulsar observations. Some prescriptions are therefore ad-hoc. We assume the magnetic field strength remains a constant when evolving the recycled pulsars back in time to obtain the initial distributions of spin periods and period derivatives. However, we allow magnetic field to decay when evolving forward in time to obtain at-merger distribution . While this inconsistency does not significantly impact our results, future models should include a self-consistent treatment of magnetic field decay.
Footnotes
- preprint: APS/123-QED
- While it is incorrect that the spin-down is due to magnetic dipole radiation, it is the dipole that plays the key role and the dipole spin-down formula is correct to order unity.
- This is used just as an example of the field evolution. Bransgrove et al. did not consider recycled pulsars in detail in their scenario (much colder crust with the field affected by accretion).
- We use the standard zero detuning high power sensitivity curve, which is publicly available at https://dcc.ligo.org/LIGO-T0900288/public. For the simplicity of calculations and computational cost with LALInference runs, we do not consider the other advanced detectors. This only makes our conclusions more conservative.
- In this work we treat the recycled NS instead of the more massive one as the primary star, so it is possible that . In this case, which corresponds to of all BNSs given the fiducial mass distributions derived in Section B, we use for the range quoted here.
- Component mass measurements were reported by R. Ferdman at the IAU Symposia 337: “Pulsar Astrophysics - The Next 50 Years” and can be found at http://pulsarastronomy.net/iaus337/post-meeting/presentations/
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