Bachelor Natuur- en Sterrenkunde (jd)
Bachelor thesis
Comparison of parameter estimates of a simulated
spinning and non-spinning Binary Neutron Star
Inspiral using TaylorF2 with 6PN, 7.5PN and NR
Tidal approximants
Ben Matthaei
27 July, 2018
Daily supervisors: Dr. Anuradha Samajdar and Prof. Chris van den Broeck Examiner: prof. dr. Frank Linde
Abstract
The rise of gravitational-wave astronomy allows physicists to investigate exotic objects such as black holes and neutron stars with new insight. Only few events have been detected so far, most notably the first Binary Black Hole merger on 14 September 2015 and the first Binary Neutron Star Merger on 17 August 2017. To make the most of the limited data available, many simulations have been run. This thesis simulated a spinning and non-spinning Binary Neutron Star inspiral system. This has been done making use of the TaylorF2 waveform model injected with a 6 PN tidal order approximant. To the waveform noise has been added and an MCMC alogrithm has been employed for parameter estimation. The recovery has been done with the 6 PN, 7.5 PN and NR tidal approximants. Results are shown as posterior plots of chirp mass, mass ratio, ˜Λ and d ˜Λ.
Author: Ben Matthaei, benji.matthaei@outlook.com UvA Student Number: 10572074
VU Student Number: 123456789
Daily supervisors: Dr. Anuradha Samajdar and Prof. Chris van den Broeck Examiner: prof. dr. Frank Linde
Second examiner: prof. dr. Auke Pieter Colijn Size: 15 EC
Submission date: 27 July, 2018
Research conducted at the National Institute for Subatomic Physics, Science Park 105, 1098 XG Amsterdam, The Netherlands between 2 April, 2018 and 15 July, 2018.
Faculty of Science
Universiteit van Amsterdam Science Park 904, 1098 XH Am-sterdam
http://www.uva.nl/fnwi/
Faculty of Science
Vrije Universiteit Amsterdam De Boelelaan 1081, 1081 HV Amsterdam
https://beta.vu.nl/
Gravitational Waves Depart-ment Nikhef
Science Park 105, 1098 XG Am-sterdam
Populair wetenschappelijke samenvatting
Zwaartekrachtsgolven zijn een natuurlijk fenomeen meer dan 100 jaar geleden voorspeld door Einstein. Pas sinds een aantal jaar is de meetapparatuur gevoelig genoeg gemaakt om de golven daadwerkelijk te detecteren. De golven worden uitgezonden door de meest compacte objecten in het universum: zwarte gaten en neutronensterren. Ondanks dat deze objecten zijn waargenomen snappen wetenschappers de natuurkunde erachter niet. Zwaartekrachts-golven zijn echter een nieuwe bron van informatie. Daarom kan het ontcijferen van de zwaartekrachtsgolven een uitermate belangrijke rol spelen in het begrijpen van deze objecten. De zwaartekrachtsgolven afkomstig van zwarte gaten en neutronensterren zijn gelijk in ba-sis, maar neutronensterren hebben een extra getijde karaktereigenschap. Om het meeste uit de nog zeer beperkte data te halen, worden simulaties gemaakt van gebeurtenissen waarbij zwaartekrachtsgolven vrijkomen. Door de simulaties te analyseren en te vergelijken met de waargenomen signalen uit de praktijk kunnen nieuwe inzichten worden verschaft. Dit onder-zoek simuleert de zwaartekrachtsgolven van twee fuserende neutronensterren. Na de golf te mengen met gesimuleerde ruis wordt getracht de eigenschappen terug te vinden. Dit wordt gedaan met een analyse die gebruik maakt van verschillende benaderingen van het getijde karakter van de golf. Zo kan de nauwkeurigheid van de verschillende getijde benaderingen worden bepaald.
Contents
1 Introduction 1 2 Theoretical background 3 2.1 TaylorF2 . . . 4 2.1.1 NRTides . . . 6 2.1.2 Parameters . . . 6 2.2 Bayesian Analysis . . . 7 3 Analysis Method 9 3.1 MCMC . . . 9 3.1.1 Burn-in period . . . 9 3.1.2 Autocorrelation time . . . 10 3.1.3 Parallel tempering . . . 10 4 Simulations 11 4.1 Choice of parameters . . . 11 5 Discussion 19 6 Conclusion 201 Introduction
After formulating the theory of general relativity in 1916, Albert Einstein predicted the existence of gravitational waves (GWs) [11]. These waves are emitted by accelerating masses, propagate at the speed of light and transport gravitational energy [12]. The gravitational energy expresses itself in the form of change in stiffness of space-time.
Since the 1960s the search for GWs began. Joseph Weber proposed the idea of using Weber bars1to detect a passing gravitational wave [23]. In a series of experiments conducted between
1965 and 1969, Weber reported on multiple occassions to have detected gravitational wave events [24, 25, 26]. However, when other scientific groups attempted to recreate the findings of Weber, making use of even more sensitive Weber bars, they all ended with no detections [21, 17, 14].
Continuing the search for GW signals, newer resonant mass detectors were developed. The second and third generation resonant mass detectors omitted the thermal noise by cooling the antenna to near zero temperatures [3]. While these detectors had an even higher sensitivity, no evidence for GWs was found.
At the same time the application of laser interferometry in the detection of GWs was being explored. The benefit of this method is an even higher sensitivity as well as the increasing bandwidth of frequencies compared to the resonant mass detectors. Currently, laser interfer-ometry is applied by using large-scale Michaelson-Morley interferometers. The idea is that when a GW passes through an arm of the interferometer, the strain on space-time resulting from the GW distorts the path to a measurable amount.
The largest ground-based laser interferometric gravitational wave observatory to date is the Laser Interferometer Gravitational-Wave Observatory (LIGO).
On September 14, 2015 a GW signal was detected for the first time by the Advanced LIGO detectors [1]. The observed signal, as seen in figure 1.1, matched the predicted waveform for two stellar-mass black holes that merged and formed a single black hole.
1
A Weber bar is a resonant mass detector operating at room temperature. Two masses are connected to each other with for example an electrostatic force. When a GW passes through this system, the masses try to follow the altering geodesic, but the connecting force will not completely allow this. The Weber bar omitted thermal noise by making use of material with a high mechanical quality factor [3].
Figure 1.1: Figure taken from the LIGO database. The figure shows a numerical relativity simulated waveform matching the reconstructed observed waveform. On the y-axis strain in the detector bands between 35-350 Hz is shown against time on the x-axis.
On August 17, 2017 both the Advanced LIGO and Advanced Virgo GW detectors detected a GW signal identified as a signal produced by a Binary Neutron Star (BNS) inspiral [2]. Approximately 1.7 seconds after the GW a short gamma-ray burst was observed by the Fermi Gamma-ray Burst monitor [16]. Upon localisation both events were found to have originated from the same source, further confirming the case of a BNS merger.
The detection of GWs gives rise to a new branch in astrophysics. As objects emitting GWs can only be directly probed from the emitted radiation, deciphering the waves can be critical to understanding the nature of black holes, as well as the equation of state for neutron stars. This thesis makes use of the LALSuite software package to simulate two BNS mergers under equal conditions, with the exception of one system consisting of spinning components and the other system consisting of non-spinning components. The BNS mergers are simulated using the analytical closed form approximant TaylorF2, detailed in section 2. The waveform will be analysed using several different order approximants with the goal of retrieving the parameters. This will be dicussed in section 3. The results will then be presented in section 4.
2 Theoretical background
To study gravitational waves, the Einstein Field Equations (EFE) are used. This is a set of 10 equations which describe gravitation as a result of spacetime being curved by mass and energy [10]. These equations have a compactly written tensor form:
Rµν−
1
2Rgµν+ Λgµν = 8πG
c4 Tµν (2.1)
Here Rµν is the Ricci curvature tensor, R the scalar curvature, gµν the metric tensor, Λ the
cosmological constant, G Newton’s Gravitational constant, c the speed of light in vacuum and Tµν the stress-energy tensor. These equations do not have an analytical solution unless
the equations are simplified. Another way to proceed is by approximating the solution. One such method is the Post-Newtonian (PN) expansion. The PN approximation computes the orbital phase evolution as a perturbative expansion in a small parameter [6]. The waveforms for Binary Black Hole (BBH) systems are already very well approximated. However for BNS systems imprints of matter lead to a more complex waveform. Where the phase of BBH systems can be described as the point-particle phase and the spin phase, BNS systems have an extra term due to tidal effects. Generally the GW can be described by the following complex waveform:
h(t) = A(t)e−iφ(t) (2.2)
A(t) is the amplitude and φ(t) the time-domain phase of the GW. Now a substitute of φ(t), the time-domain phase, to φ(ˆω), the dimensionless GW phase can be made, where ˆω is the dimensionless GW frequency given by ˆω = M ∂tφ(t). Here M is the total mass of the binary.
For BNS systems the following PN-inspired phasing formula can be made, constructed of three individual contributions:
φ(ˆω) ≈ φpp(ˆω) + φspin(ˆω) + φtides(ˆω) (2.3)
The first two terms are the same terms used to describe a BBH, these are the non-spinning point particle phase φppand the spinning phase term, φspin, including spin effects. The third
term, φtides, incorporates the tidal contribution [13].
Each of these contributions can be estimated using the PN approximation. The PN ex-pansion is expressed as a sum and the order of correction is expressed as two times the PN
order. Several studies have been dedicated to add higher order corrections to this expansion. Currently the point particle phase term has been approximated upto an order of 3.5PN [5]. The tidal phase term comes in at a leading order of 5PN and has been approximated upto an order of 7.5 PN [8].
With the PN approximation a waveform’s phase can be calculated, which is important as a GW detector gets a constant stream of data. This data is made up of both signal and noise, which can be represented as
x(t) = h(t) + n(t) (2.4)
x represents the detector output, h the signal and n the noise. Signals from Compact Binary Coalescences (CBCs) are extremely well modelled. However it is challenging to extract such a signal buried in detector noise with arbitrary parameters. A method called matched filtering [20] is therefore employed to verify the presence of such a signal in the data. The use of matched filtering requires the template to be well known. Hence it is important to have accurate approximants in order to verify a signal.
2.1 TaylorF2
Different PN waveform templates are described by Buonanno et al. [6]. One in particular is called the TaylorF2 waveform. This template can generate signals emitted by inspiraling compact binaries. To evaluate signals using this template the time domain signal is Fourier transformed to the frequency domain. As the full time (or frequency spectrum) has to be evaluated, the signal will be in the form of an integral. To solve this integral the stationary phase approximation (SPA) is used. The SPA is a method to approximate an integral by analytically solving the integrand in a domain where it’s contribution is largest. Using the SPA the waveform in the frequency domain takes the following form:
˜ B(f ) =qA(tf) ˙ F (tf) ei[ψf(tf)−π4], ψf(t) ≡ 2πf t − 2φ(t) dφ dt ≡ 2πF (t) (2.5)
In this equation A is the amplitude of the waveform, ˙F the time derivative of the GW frequency and φ is the phasing formula. The time tf is defined as the time where the Fourier
variable f is equal to the GW frequency. This method on the 3.5PN phase approximation given by Blanchet et al. [5] has been calculated by Damour et al. [7] who arrive at the following form:
˜
h(f ) = Af−76e iψ(f )
Where A ∝ M 5 6Q D where M = (m1m2) 3 5(m1+ m2)− 1
5 is the chirp mass, Q the variable for
the angles and D the distance to the binary. To the order of 3.5PN the waveform is given by:
ψ(f ) ≡ ψf(tf) − π 4 = 2πf tc− φc− π 4 + 3 128ηv5 N X k=0 αkvk (2.7)
tc is the coalescence time, φc the phase at the time of coalescence, η the symmetric mass
ratio and the characteristic velocity v = (πM f )13 with M = m1 + m2 the total mass of the
binary. The coefficients αk have been calculated to the 7-th order (3.5PN) and are given by:
α0 = 1 (2.8a) α1 = 0 (2.8b) α2 = 20 9 743 336+ 11 4 η (2.8c) α3 = −16π (2.8d) α4 = 10 3058673 1016064 + 5429 1008η + 617 144η 2 (2.8e) α5 = π 38645 756 + 38645 252 log v vlso −65 9 η h 1 + 3log v vlso i (2.8f) α6 = 11583231236531 4694215680 − 640π2 3 − 6848γ 21 + η − 15335597827 3048192 + 2255π2 12 − 1760θ 3 + 12320λ 9 +76055 1728 η 2−127825 1296 η 3−6848 21 log(4v) (2.8g) α7 = π 77096675 254016 + 378515 1512 η − 74045 756 η 2 (2.8h)
Equations and coefficients are taken from [4].
However the calculation for the phase so far only describes the spinning point-particle. In order to describe a BNS inspiral, the tidal contribution to the phasing has to be added. These terms come into play at an order of 5 PN.
ψ(f ) = ψP P(f ) + ψtidal(f ) (2.9)
The tidal phasing term has been calculated by Damour et al. [8] to the 15-th order (7.5PN). The tidal phasing term is given by:
ψtidal(f ) = 3 128ηv −5 2 X A=1 λA M5X A − 24(12 − 11XA)v10 + 5 28(3179 − 919XA− 2286X 2 A+ 260XA3)v12+ 24π(12 − 11XA)v13 − 2439927845 508032 − 480043345 9144576 XA+ 9860575 127008 X 2 A −421821905 2286144 X 3 A+ 4359700 35721 X 4 A− 10578445 285768 X 5 A v14 + π 28(27719 − 22127XA+ 7022X 2 A− 10232x3A)v15 (2.10)
Here XA= mMA, λA= λ(mA) and A = 1, 2 denotes the subscript of the binary component.
With these phasing terms, the TaylorF2 template can create a waveform given a set of parameters.
2.1.1 NRTides
Another approach to find solutions to the EFE is by using Numerical Relativity (NR). This method simulates the physics of a general relativistic problem to gather data. Such a simu-lation is very time intensive, yet very accurate. Dietrich et al. [9] used an NR simusimu-lation to create an approximant for the tidal phasing and added this to already existing NR models of BBH systems to gain a resulting waveform of a BNS system. The NR tidal phasing term in the frequency domain is given by:
ΨN RT idal(f ) = −κTef fc˜N ewtx 5 2 XAXB PΨN RT idal PΨN RT idal= 1 + ˜n1x + ˜n3 2x 3 2 + n2x2+ n5 2x 5 2 + n3x3 1 + d1x + d3 2x 3 2 κTef f = 2 13 h 1 + 12XB XA XA CA 5 kA2 + (A ↔ B)i (2.11)
ΨN RT idalgives the NR approximation for the tidal phase where PΨN RT idaldenotes a coeffi-cient and κTef f is the tidal parameter connected to Λ1 and Λ2.
2.1.2 Parameters
To simulate a gravitational wave, the waveform will need a set of parameters. These include at least 9 parameters for non-spinning, circular binaries made of point-mass objects [22]:
• The component masses m1 and m2, which are often expressed as the chirp mass, M
= (m1m2)
3
5(m1+ m2)− 1
5, the mass ratio, q = m2
m1 and the symmetric mass ratio, η =
m1m2
• The luminosity distance to the source dL.
• The right ascension, α and declination δ. These variables indicate the location in the sky as seen from the detector.
• The inclination angle ι. This angle indicates the tilt between the system’s orbital plane and the line of sight.
• The polarisation angle ψ describing the orientation of the binary’s orbital momentum vector onto the plane on the sky.
• A reference time tc which is arbitrary. It can be chosen as the moment at which the binary merges.
• The binary’s orbital phase φcat the reference time tc.
As BNS waveforms have matter imprints, an additional two parameters are needed to describe a non-spinning BNS:
• The tidal deformability, Λ1 and Λ2, of both neutron stars. This is the characteristic
parameter describing the tidal susceptibility of the compact objects. Generally for spinning binaries another six parameters are needed:
• The dimensionless spin magnitudes χi defined as χi ≡ ~ |si|
m2
i. Here si is the spin vector of
the object i.
• For every si there are two angles specifying the orientation with respect to the plane
defined by the line of sight and the initial orbital angular momentum.
However TaylorF2 is an aligned spinning model therefore just two additional parameters are needed which are the magnitude of both spins.
LIGO-Virgo data analysis is carried out using the open-source package LALSuite, which forms a part of the Ligo Algorithm Library (LAL). Data analysis has a particular focus on the waveform module LALSimulation, which is the library used for simulating available waveform models. For parameter estimation purposes however, all the analysis find it’s core at Bayesian inference, for which the module LALinference is used.
2.2 Bayesian Analysis
Bayesian analysis on GW signals approaches the data from two sides, model selection and parameter estimation. Model selection chooses the most probable waveform according to the
observed data, while parameter estimation uses the observed data under a fixed waveform model to estimate the parameters. Both problems can be explored using Bayesian inference but this thesis only makes use of parameter estimation. To use Bayesian inference Bayes’ Theorem is needed.
Bayes’ Theorem estimates an event as probability taking prior knowledge of conditions, which might be related to the event, into account. Mathematically Bayes’ Thereom takes the following form:
P (A|B) = P (B|A)P (A)
P (B) (2.12)
In this equation A and B are events. P (A|B) indicates the probability of event A happening if B is true. Likewise P (B|A) indicates the probability of event B happening if A is true. P (A) and P (B) are the probabilities of events A and B. Bayesian inference uses Bayes’ Theorem to derive a posterior probability distribution. In equation 2.12 Bayesian inference substitutes the single events A and B by a set of events. For parameter estimation an unknown parameter can be represented as a probability density p(θ|H), whereR p(θ|H) = 1. Here θ stands symbol for the full collection of parameters and H is the hypothesis. Applying Bayesian inference the following formula can be written:
p(θ|d, H) = p(θ|H)p(d|θ, H)
p(d, H) (2.13)
Parameter estimation is done by providing a prior probability distribution p(θ|H) which is then updated upon receiving new data d to estimate a posterior probability distribution p(θ|d, H) [22].
3 Analysis Method
With the TaylorF2 template a waveform can be created. To resemble a realistic datastream noise is added to this waveform. This noise has a stationary, gaussian form. The goal is to extract the parameters from this data. For this an algorithm is needed.
3.1 MCMC
The algorithm used for the parameter estimation is a Markov Chain Monte Carlo (MCMC) algorithm. MCMC is a process that stochastically walks through parameter space (Monte Carlo) and adds parameters to the posterior based on the prior which is updated upon re-ceiving a new parameter (Markov Chain). To use formula 2.13 a proposal density function is required. For this a uniform distribution is chosen between boundary values which are based on an educated guess. In this uniform distribution a stochastically chosen parameter θ is compared to a comparison parameter θ0 under given evidence. Using Bayes’ Theorem the likelihood of both parameters can be calculated as a fraction of each other
α = Q(θ|θ
0)p(θ|d, H)
Q(θ0|θ)p(θ|d, H) (3.1)
Formula 3.1 expresses the likelihood of θ against the likelihood of θ0. If α > 1, the new parameter θ is more likely than the comparison parameter θ0. In this case θ is added to the chain and replaces θ0 as the new comparison parameter. If α ≤ 1, θ is rejected and θ0 remains the comparison parameter. The posterior distribution will consist of the accepted parameters. If the hypothesis used in Bayes’ Theorem is correct, then with a sufficient amount of iterations the posterior distribution will peak at the parameter value searched for.
3.1.1 Burn-in period
As the first parameter, which becomes the comparison parameter, is also stochastically chosen, it might be a very unlikely parameter. The next tested parameter might also be very unlikely, but still more likely than the comparison parameter. This means that it will be added to the chain and will falsely represent the posterior that is being searched for. This time is called the burn-in period. To omit this, a fixed number of samples are discarded when estimating
the posterior. As the process adds more iterations, not finding samples from the posterior becomes highly unlikely.
3.1.2 Autocorrelation time
Another problem which has to be resolved is that parameters that lie closely together in parameter space are typically correlated. If a large number of close lying parameters are added to the chain this can also falsely represent the posterior. An autocorrelation time (ACT) τ is defined as
τ = 1 + 2X
t
ˆ
c(t) (3.2)
t is the iteration and ˆc is the Pearson correlation coefficient between the chain of samples and itself shifted by t samples. Only every τ -th sample is used to thin the chain.
3.1.3 Parallel tempering
To get a more accurate estimation of the posterior, multiple chains are started in a process called parallel tempering. As all chains are expected to converge to the most likely value after an adequate number of iterations, the relative distance between these chains can be calculated as a measure of accuracy. This is done with the Gelman-Rubin ˆR [15]. For the purpose of parameter estimation if ˆR < 1.01 the chains are considered to have converged.
4 Simulations
Now that a waveform model is given and a parameter estimation algorithm is outlined, all that is left to do is defining a set of parameters to create the waveform which can be analysed.
4.1 Choice of parameters
In section 2.1.2 the parameter arguments of the TaylorF2 function are outlined. The two BNS inspiral systems simulated in this thesis use the same parameters, with the exception of spin.
• For m1 and m2 masses of m1 = 1.4M and m2 = 1.35M were chosen, where M ≈
2 ∗ 1030 kg, is the solar mass. The range of neutron star masses is currently unknown. While theoretical lower limits go to masses of ≈ 0.1M , the smallest mass detected of
a neutron star is ≈ 1.174M [19]. Theoretical upper limits go to masses of ≈ 3M
and the largest detected neutron star has a mass of ≈ 2.27M [18]. A typical mass
for a neutron star is ≈ 1.4M which has been chosen for m1. To avoid a system with
symmetric masses m2 was chosen slightly smaller.
• For dL a distance of 50 Mpc was chosen and for the inclination ι an angle of 0◦ has been chosen. These parameter together directly affect the observed amplitude. For this thesis these parameters are of minor importance and their only bound is to be chosen in a combination that allows the observed signal to be strong enough for a clear detection. • For α and δ angles of 60◦ have been chosen. For the polarisation an angle of 0◦ has
been chosen. These 3 parameters have an effect on how much of the incoming amplitude can be measured by the detector. For this thesis these parameters are also of minor importance and are only bound to a combination for which the signal is strong enough to be clearly detected.
• A reference time, tc, which is arbitrary. It can be chosen as the time as the moment at
which the binary merges.
• For the tidal deformabilities Λ1 and Λ2 there is a direct relation to the component
masses. As component masses were already chosen Λ1 and Λ2 have been set to 400 and
500 respectively.
The spinning binary has an additional two parameters:
• Both spin components are chosen along the z-axis with a magnitude of 0.4.
With these parameters the following waveforms were created using the TaylorF2 template with Tidal approximants of 6PN, 7.5PN and NR. Figure 4.1 shows the waveforms of the spinning and non-spinning BNS systems. On the y-axis the strain can be seen against the time on the x-axis.
Figure 4.1: From top to bottom the spinning and non-spinning BNS system waveforms created with TaylorF2 using tidal approximants of 6 PN (blue), 7.5 PN (red) and NR
A second set of waveforms, see figure 4.2, has been created with no tidal deformabilities and approximated with the 6 PN, 7.5PN and NR tidal approximants. This is done to make sure the tidal approximants do not have an effect on the base waveform.
Figure 4.2: From top to bottom the spinning and non-spinning waveforms without tidal de-formabilities created with TaylorF2 using tidal approximants of 6 PN (blue), 7.5 PN (red) and NR (green). The different approximants perfectly overlap as ex-pected as the different approximants estimate the tidal phasing, which is excluded by removing the tidal deformabilities.
From figure 4.1 the 6 PN (blue) waveforms were used and noise was added. Figure 4.3 illustrates the noise compared to the signal by plotting the strain caused by noise and strain caused by a GW signal in one figure. Subsequently the waveforms were subjected to the MCMC algorithm described in section 3 and the 6 PN, 7.5 PN and NR waveforms were used for the recovery. For both runs some interesting recovered parameters have been plotted. These are the chirp mass, M, the mass ratio q, ˜Λ and d ˜Λ. ˜Λ and d ˜Λ are linear combinations of the tidal deformabilities and masses of the binary components.
M = (m1m2) 3 5(m1+ m2)− 1 5 (4.1a) q = m2 m1 (4.1b) ˜ Λ = 16 13 (m1+ 12m2)m41Λ1+ (m2+ 12m1)m42Λ2 (m1+ m2)5 (4.1c) d ˜Λ = 1 2 hp 1 − 4η 1 −13272 1319η + 8944 1319η 2(Λ 1+ Λ2) +1 −15910 1319η + 32850 1319η 2+3380 1319η 3(Λ 1− Λ2) i (4.1d)
The posterior plots are normalised and show the probability distributions found by the differ-ent tidal approximants. For reference the injected values have been plotted as vertical dashed lines.
Figure 4.3: Comparison plot of the a GW signal (red) and detector noise (blue). This figure displays that the strain caused by noise is considerably larger than the strain caused by the GW signal.
Figure 4.4: The normalised posterior distributions for the non-spinning BNS recovered with 6 PN (blue), 7.5 PN (green) and NR (red). From top to bottom and left to right, the chirp mass mc, mass ratio q, ˜Λ and d ˜Λ are shown. The dashed vertical lines represent the injected parameters.
Figure 4.4 shows the recovered normalised posteriors for the non-spinning BNS system. All approximants have recovered the chirp mass (top left) very well as the peak lies very near the injected value. For q (top right) the injected value is quite far from the peak, however the peak seems to be consistent among the different approximants. ˜Λ (bottom left) is closest approximated with the 7.5 PN approximant, notable is the peak of the NR approximant, as it lies furthest from the injected value yet has the smallest deviation. For d ˜Λ (bottom right) the NR approximant shows the most accurate result with a clear peak converging to the injected value. While the PN approximants also seem to be converging to the injected value, their recovery is less accurate with the area between values of -100 and 100 showing resemblence to a uniform distribution.
Figure 4.5: The normalised posterior distributions for the spinning BNS recovered with 6 PN (blue), 7.5 PN (green) and NR (red). From top to bottom and left to right, the chirp mass mc, mass ratio q, ˜Λ and d ˜Λ are shown. The dashed vertical lines represent the injected parameters.
In figure 4.5 the recovered normalised posteriors for the spinning BNS system are shown. Like the non-spinning case the chirp mass (top left) is very well recovered. However q (top right) is less well recovered, where the area between 0.5 and 1 for both PN approximants have some resemblance to a uniform distribution. Every approximant also shows a peak between 0.2 and 0.3 especially notable on the NR approximant. All approximants show a double peak for ˜Λ (bottom left) most notable on the NR approximant as both peaks have approximately the same size. For the PN approximants the injected value seems to lie between the peaks, near the local minimum. d ˜Λ (bottom right) seems to have recovered well on all approximants with clear peaks on the injected value.
5 Discussion
The results of figure 4.4 and 4.5 show a couple of peculiarities. While both posteriors of the chirp mass M have recovered well with all approximants, both mass ratios have not recovered well with any approximant. These variables share a degeneracy as both are functions of m1
and m2 and an increase of certainty in one will increase the uncertainty in the other.
While the NR tidal approximant is expected to give the most sophisticated results, the posterior of ˜Λ of the non-spinning case (figure 4.4 bottom left) shows the opposite. Curious enough the NR approximant has a peak with the smallest spreading and something might have induced a translation along the x-axis. The posterior of ˜Λ for the spinning BNS shows two peaks on every approximant. This double-peak on the posterior of ˜Λ might be induced by spin effects as the posterior of ˜Λ for the non-spinning BNS shows only one peak for every approximant. Investigations of the double-peak of ˜Λ of the spinning BNS and why the approximants recovered a minimum at the injected value are in progress.
6 Conclusion
A spinning and non-spinning BNS inspiral system were simulated using the TaylorF2 wave-form with 6 PN. The recovery made use of the 6 PN, 7.5 PN and NR tidal approximants. Results of chirp mass, mass ratio, ˜Λ and d ˜Λ are plotted as posterior distributions. In both cases the chirp mass has been recovered very well with all approximants. In both cases all approximants show a similar curve for the mass ratio but the estimated posterior shows the injected value is not very likely. A possible explanation is the degeneracy the chirp mass and the mass ratio share. ˜Λ of the non-spinning binary has been estimated best by the 7.5 PN approximant. While it is expected for the NR approximant to give the most refined results it lies furthest from the injected value but also shows the smallest deviation in it’s estima-tion. Because of the small deviation, the NR approximation appears to be accurate despite lying far away from the injected value and investigation into whether something is causing a translation is needed. ˜Λ of the spinning binary shows two peaks on the posterior of every approximant. As the ˜Λ of the non-spinning BNS does not show double-peaks on the approx-imants it is possibly spin related. To find out whether the two peaks of the posterior are spin related or an anomaly more simulations have to be run making use of different algorithms for the parameter estimation.
Acknowledgements
I would like to thank dr. Anuradha Samajdar for guiding me through this project and always taking the time to answer all my questions. I am grateful for all the time you have invested in me and the patience you have shown explaining the difficult topics of this field. I have learned a lot from you these past few months and despite the short duration of the project it has been a very informative and enjoyable experience. I would also like to thank prof. dr. Chris van den Broeck for having me at the gravitational waves department of Nikhef. Thank you both for trusting me with this project.
I would like to thank prof. dr. Frank Linde for helping me find a project despite initially not offering a project and also for becoming my supervisor. I also would like to thank prof. dr. Auke-Pieter Colijn for agreeing to be my second examiner.
Further I would like to thank my father, Bob Matthaei, with whom I’ve often reflected throughout the project to gain a better understanding of what I have been doing. Last but not least I would like to thank my mother, Lita Lie, and my partner, Gooitske Fennema, for always having their unconditional support.
Bibliography
[1] B. P. e. a. Abbott. Observation of gravitational waves from a binary black hole merger. Phys. Rev. Lett., 116:061102, Feb 2016. doi: 10.1103/PhysRevLett.116.061102. URL https://link.aps.org/doi/10.1103/PhysRevLett.116.061102.
[2] B. P. e. a. Abbott. Gw170817: Observation of gravitational waves from a binary neutron star inspiral. Phys. Rev. Lett., 119:161101, Oct 2017. doi: 10.1103/PhysRevLett.119. 161101. URL https://link.aps.org/doi/10.1103/PhysRevLett.119.161101. [3] O. Aguiar. The past, present and future of the resonant-mass gravitational wave
detec-tors. 2010. doi: 10.1088/1674-4527/11/1/001. URL https://arxiv.org/ftp/arxiv/ papers/1009/1009.1138.pdf.
[4] K. G. Arun, B. R. Iyer, B. S. Sathyaprakash, and P. A. Sundararajan. Parameter estimation of inspiralling compact binaries using 3.5 post-Newtonian gravitational wave phasing: The nonspinning case. 71(8):084008, Apr. 2005. doi: 10.1103/PhysRevD.71. 084008.
[5] L. Blanchet, G. Faye, B. R. Iyer, and B. Joguet. Gravitational-wave inspiral of compact binary systems to 7/2 post-newtonian order. Phys. Rev. D, 65:061501, Feb 2002. doi: 10.1103/PhysRevD.65.061501. URL https://link.aps.org/doi/10.1103/PhysRevD. 65.061501.
[6] A. Buonanno, B. R. Iyer, E. Ochsner, Y. Pan, and B. S. Sathyaprakash. Comparison of post-Newtonian templates for compact binary inspiral signals in gravitational-wave detectors. 80(8):084043, Oct. 2009. doi: 10.1103/PhysRevD.80.084043.
[7] T. Damour, B. R. Iyer, and B. S. Sathyaprakash. Comparison of search templates for gravitational waves from binary inspiral. Phys. Rev. D, 63:044023, Jan 2001. doi: 10. 1103/PhysRevD.63.044023. URL https://link.aps.org/doi/10.1103/PhysRevD.63. 044023.
stars in late-inspiral gravitational-wave signals. 85(12):123007, June 2012. doi: 10.1103/ PhysRevD.85.123007.
[9] T. Dietrich, S. Khan, R. Dudi, S. J. Kapadia, P. Kumar, A. Nagar, F. Ohme, F. Pan-narale, A. Samajdar, S. Bernuzzi, G. Carullo, W. Del Pozzo, M. Haney, C. Markakis, M. Puerrer, G. Riemenschneider, Y. Eka Setyawati, K. W. Tsang, and C. Van Den Broeck. Matter imprints in waveform models for neutron star binaries: tidal and self-spin effects. ArXiv e-prints, Apr. 2018.
[10] A. Einstein. Die Grundlage der allgemeinen Relativit¨atstheorie. Annalen der Physik, 354:769–822, 1916. doi: 10.1002/andp.19163540702.
[11] A. Einstein. Nherungsweise integration der feldgleichungen der gravitation. Sitzungs-berichte der Kniglich Preuischen Akademie der Wissenschaften Berlin, pages 688–696, 1916.
[12] A. Einstein. ber gravitationswellen. Sitzungsberichte der Kniglich Preuischen Akademie der Wissenschaften Berlin, pages 688–696, 1918.
[13] ´E. ´E. Flanagan and T. Hinderer. Constraining neutron-star tidal Love numbers with gravitational-wave detectors. 77(2):021502, Jan. 2008. doi: 10.1103/PhysRevD.77. 021502.
[14] R. L. Garwin and J. L. Levine. Single gravity-wave detector results contrasted with previous coincidence detections. Phys. Rev. Lett., 31:176–180, Jul 1973. doi: 10. 1103/PhysRevLett.31.176. URL https://link.aps.org/doi/10.1103/PhysRevLett. 31.176.
[15] A. Gelman and D. B. Rubin. Inference from iterative simulation using multiple sequences. Statist. Sci., 7(4):457–472, 11 1992. doi: 10.1214/ss/1177011136. URL https://doi. org/10.1214/ss/1177011136.
[16] A. e. a. Goldstein. An ordinary short gamma-ray burst with extraordinary implications: Fermi-gbm detection of grb 170817a. Astrophys. J., 848, 2017.
[17] J. L. Levine and R. L. Garwin. Absence of gravity-wave signals in a bar at 1695 hz. Phys. Rev. Lett., 31:173–176, Jul 1973. doi: 10.1103/PhysRevLett.31.173. URL https: //link.aps.org/doi/10.1103/PhysRevLett.31.173.
[18] M. Linares, T. Shahbaz, and J. Casares. Peering into the dark side: Magnesium lines establish a massive neutron star in psr j2215+5135. The Astrophysical Journal, 859(1): 54, 2018. URL http://stacks.iop.org/0004-637X/859/i=1/a=54.
[19] J. G. Martinez, K. Stovall, P. C. C. Freire, J. S. Deneva, F. A. Jenet, M. A. McLaughlin, M. Bagchi, S. D. Bates, and A. Ridolfi. Pulsar J0453+1559: A Double Neutron Star System with a Large Mass Asymmetry. 812:143, Oct. 2015. doi: 10.1088/0004-637X/ 812/2/143.
[20] B. S. Sathyaprakash and S. V. Dhurandhar. Choice of filters for the detection of grav-itational waves from coalescing binaries. Phys. Rev. D, 44:3819–3834, Dec 1991. doi: 10.1103/PhysRevD.44.3819. URL https://link.aps.org/doi/10.1103/PhysRevD. 44.3819.
[21] J. A. Tyson. Null search for bursts of gravitational radiation. Phys. Rev. Lett., 31:326– 329, Jul 1973. doi: 10.1103/PhysRevLett.31.326. URL https://link.aps.org/doi/ 10.1103/PhysRevLett.31.326.
[22] J. e. a. Veitch. Parameter estimation for compact binaries with ground-based gravitational-wave observations using the lalinference software library. Phys. Rev. D, 91:042003, Feb 2015. doi: 10.1103/PhysRevD.91.042003. URL https://link.aps.org/ doi/10.1103/PhysRevD.91.042003.
[23] J. Weber. Detection and generation of gravitational waves. Phys. Rev., 117:306–313, Jan 1960. doi: 10.1103/PhysRev.117.306. URL https://link.aps.org/doi/10.1103/ PhysRev.117.306.
[24] J. Weber. Gravitational radiation. Phys. Rev. Lett., 18:498–501, Mar 1967. doi: 10. 1103/PhysRevLett.18.498. URL https://link.aps.org/doi/10.1103/PhysRevLett. 18.498.
[25] J. Weber. Gravitational-wave-detector events. Phys. Rev. Lett., 20:1307–1308, Jun 1968. doi: 10.1103/PhysRevLett.20.1307. URL https://link.aps.org/doi/10.1103/ PhysRevLett.20.1307.
[26] J. Weber. Evidence for discovery of gravitational radiation. Phys. Rev. Lett., 22:1320– 1324, Jun 1969. doi: 10.1103/PhysRevLett.22.1320. URL https://link.aps.org/doi/ 10.1103/PhysRevLett.22.1320.
