University of Amsterdam

MSc Astronomy & Astrophysics Gravitation AstroParticle Physics Amsterdam Master Thesis

Reliability of the Parameterised Test of General Relativity on GW150914 and GW151226 An assessment of model waveforms and robustness of the test’s results

by Janna Goldstein 10186832 August 2016 60 ECTS September 2015 - July 2016

Supervisor: Dr. Chris van den Broeck

Examiners: Dr. Chris van den Broeck & Dr. Dorothea Samtleben

National Institute for Subatomic Physics

Summary The parameterised test of general relativity (GR) is a generic test of GR that uses a Bayesian approach to constraining parameterised deviations. It uses gravitational waves from coalescing compact binaries such as the two signals observed by the LIGO detectors in the past year, which consist of an inspiral stage, merger and ringdown of the final black hole. The analysis depends on the accuracy of the phenomenological gravitational waves model IMRPhenomPv2 that produces a full inspiral-merger-ringdown waveform. The model can only be directly verified using the limited number of numerical relativity waveforms that have been produced. In this project, the accuracy of IMRPhenomPv2 is studied more extensively by comparing it with the different, quasi-analytical model SEOBNRv3, which can be done for any choice of binary parameters. This comparison is done by calculating the faithfulness F, a measure of the overlap of the two waveforms. Around the parameters of both events GW150914 and GW151226, it was found that the models are mostly in good agreement, with F ≥ 0.96, and that lower matches can often be explained by a lack of calibration numerical relativity waveforms used in constructing the models. It was also found that generally, the stronger the precession effects in the spins of the black holes, the lower the faithfulness of the two models. Additionally, the results of the parameterised test of GR on GW150914 are investigated in two tests. Seemingly, deviations from GR of 2–2.5σ are found for the inspiral region, which is suspected to be due to fluctuations in the noise. This is investigated by running the parameterised test on fifteen stretches of detector noise, in which a numerical relativity waveform is injected. The results are deemed inconclusive as the recovered signal to noise ratio of most runs is much higher than that of GW150914, however offsets are found at 1–2σ. A second test uses injections of model waveforms (IMRPhenomPv2) with built-in deviations from GR, to probe the sensitivity of the parameterised test. It was found that deviations in the inspiral regime are recovered somewhat less significantly than the injected deviations, which might be due to the previously mentioned noise effect. Some deviations in the merger-ringdown regime are measured very strongly, while others are not, indicating a possibly varying sensitivity of the parameterised test.

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Contents Summary

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Acknowledgements

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Introduction

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1 Gravitational waves 1.1 Black hole binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Detections of gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 6 7

2 Testing general relativity 2.1 The strong-field regime . . . . . . . . . . . . . 2.2 The parameterised test of GR . . . . . . . . . 2.2.1 The nested sampling algorithm . . . . 2.2.2 Results for GW150914 and GW151226

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3 Waveforms 3.1 Waveform approximations . . . . 3.1.1 Effective One Body . . . . 3.1.2 Phenomenological models 3.2 Overlaps . . . . . . . . . . . . . . 3.2.1 Faithfulness . . . . . . . . 3.2.2 Probing parameter space 3.2.3 Results . . . . . . . . . . 3.2.4 Discussion . . . . . . . . .

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parameterised test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Reliability and sensitivity of the 4.1 The effect of noise . . . . . . . 4.1.1 Results . . . . . . . . . 4.1.2 Discussion . . . . . . . . 4.2 Sensitivity of the parameterised 4.2.1 Results . . . . . . . . . 4.2.2 Discussion . . . . . . . .

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Conclusions

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A Results for the noise investigation

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B Results for the non-GR waveforms

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Acknowledgements I would like to sincerely thank Dr. Chris van den Broeck, my supervisor, for his guidance, help and comments on my work during a busy time for gravitational waves. My gratitude extends as well to Dr. Dorothea Samtleben, for filling the role of examiner. Special thanks to Jeroen Meidam for all his help and patience. I have greatly appreciated being part of the gravitational waves group of Nikhef; thank you all. Finally, I wish to express my gratitude towards my friends and family for their support.

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Introduction The detection of gravitational waves from the merger of a binary black hole made by the LIGO Virgo Collaboration on the 14th of September 2015, GW150914 (Abbott et al., 2016b), has heralded a new era of gravitational wave astronomy. Together with the second detection GW151226 (Abbott et al., 2016a), also of two merging black holes, this shows that it is now possible to observe events that are hardly, or perhaps not at all, visible with electromagnetic telescopes. Given the distance to which the most energetic events can be seen, this provides an exciting prospect for both astrophysics and cosmology. From a physics perspective, gravitational waves are interesting because they probe strong, dynamical gravity. Thus, they provide the first possibility to test general relativity (GR) beyond the weak field regime. Such a test is implemented in the parameterised test of GR, discussed in Chapter 2, which has been developed in the gravitational waves group of Nikhef (National Institute of Subatomic Physics). In this work, several aspects of the reliability of the test’s methods and results are studied, focusing on its application to GW150914 and GW151226. The parameterised test is a search for generic deviation from GR in signals from coalescing binaries, consisting of neutron stars and/or black holes. In order to do so, deviation parameters are introduced in analytical waveforms that are approximations of gravitational waves as predicted by GR. These approximants are needed as it takes too long to calculate a large number of waveforms precisely, since this can only be done numerically. Naturally, it is necessary to study the reliability of these approximants, which is addressed in Chapter 3 of this thesis. Chapter 4 is an assessment of the robustness and sensitivity of the parameterised test; how much do random noise fluctuations affect the result? And how strong must a deviation from GR be to be picked up? Leading up to all this, the first chapter will provide an introduction on gravitational waves, specifically those from binary black hole mergers, and the detections GW150914 and GW151226.

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Chapter 1

Gravitational waves Gravitational waves were first predicted by Einstein (1916, 1918) as a vacuum solution to the equations of general relativity (GR): 1 8πG Rµν − gµν R = 4 Tµν , (1.1) 2 c where gµν is the metric, Rµν is the Ricci curvature tensor, R the Ricci scalar, G the Newtonian gravitational constant, c the speed of light and Tµν the stress-energy tensor. These equations are highly non-linear and can not be solved exactly, save for a few special cases (for example, a single black hole). It is possible to simulate GR spacetime, which is called numerical relativity. However, far away from any strong curvature, for example at the site of a gravitational waves detector, the equations can be linearised. This is done by introducing a small perturbation h on a background Minkowski metric: gµν = ηµν + hµν , |h|  1.

(1.2)

Furthermore, the detector site is considered a vacuum, so Tµν = 0. With this the Einstein equations can be rewritten, using some gauge freedom as well, to obtain: hµν = 0, or with the d’Alambertian written out:  ∂2 ∂x2

+

∂2 ∂2  1 ∂2 + hµν , h = µν ∂y 2 ∂z 2 c2 ∂t2

which is a wave equation. The speed of these gravitational waves (GW) is c, the speed of light. The solutions are plane waves, and can be written (in the transverse-traceless gauge), as:   h× hm + 0 hT T = h+ −hm × 0 cos(ω(t − z/c)). 0 0 0

(1.3)

(1.4)

(1.5)

Here, the wave is traveling in the z-direction, and has an angular frequency ω. h+ and h× are the two polarisations of the wave. The effect of a wave with only non-zero h+ is to expand and compress space in the horizontal and vertical directions of the plane, perpendicular to the propagation direction. For a non-zero h× , the same thing happens under a rotation of 45◦ . The shape of the polarisations depends on the source of the gravitational waves, which can be anything with a mass quadrupole moment, or higher multipole moment (there is no gravitational mono- or dipole radiation). Therefore, even though the waves near a detector are described in linearised gravity, they do contain full non-linear GR from the source. This is what can be measured with GW detectors. Since gravity is such a weak force, these waves are generally incredibly tiny. Any detectable waves must come from a source with very strong spacetime curvature, and thus from very massive and compact objects with a rapidly varying quadrupole (or higher multipole) moment. 5

Figure 1.1: Model (template) waveform and numerical relativity waveform constructed using the estimated parameters of GW150914. On top, a schematic picture of the binary black hole coalescence is plotted, each stage corresponding to the region of the waveform below. Picture taken from Abbott et al. (2016b).

1.1

Black hole binaries

Compact binaries, consisting of two neutron stars and/or black holes, are among the most massive and compact sources of gravitational waves (GW) possible, and GWs from merging binary black holes are the only ones detected so far (Abbott et al., 2016b). In a compact binary, the orbit of the stars creates a mass quadrupole moment, such that GWs carry away energy. Thus, the orbit decays into a closer orbit with a more rapidly changing quadrupole, such that stronger GWs are emitted and so on, until eventually the objects merge. The decrease in the orbital period was famously observed in the Hulse-Taylor binary (Hulse and Taylor, 1975; Taylor and Weisberg, 1982), consisting of a neutron star and a pulsar. It is, however, far from the point of merger. On the other hand, binaries that are observed with gravitational waves are in the last stages of the inspiral, where the waves’ amplitude and frequency are high enough to be detectable by laser interferometers such as LIGO. The waves can be observed through the merger of the binary, after which there is an additional stage called the ringdown. This ringdown is caused by the damped oscillations of the final black hole that was created during merger. A schematic of the inspiral-merger-ringdown and the corresponding waveform is shown in Fig. 1.1, as given by Abbott et al. (2016b). There is no exact solution for the waves from compact binaries, but an excellent approximation for the inspiral stage is made by expanding the equations in powers of v/c, where v is the characteristic velocity of the binary and c the speed of light. With Keplers third law, this velocity is given by: v = (πGM fGW )1/3 ,

(1.6)

where fGW is the gravitational wave frequency, which is twice the orbital frequency. Such an expansion is called a post-Newtonian approximation, or PN for short. Due to a historical mishap, the PN orders are indicated with half-integers, referring to the power in v/c by twice as much; for example, 1.5PN would refer to the coefficients multiplying (v/c)3 beyond leading order. It is assumed that the orbital frequency does not change significantly over the course of one orbit, which 6

is false only for the last stages of the inspiral. In this adiabatic approximation, the system can be described with two differential equations: 3 ˙ = v , φ(t) (1.7a) GM −F (v) . (1.7b) v˙ = 0 E (v) ˙ which is obtained by setting it equal to The first equation gives the time derivative of the orbital phase φ, the orbital angular frequency ωorbit = πfGW , and rewriting fGW using the expression for v in Eq. (1.6). The second equation simply equates the rate of binding energy E lost with the energy flux of the gravitational ∂E ∂v dE = = E 0 v. ˙ Both the flux F and the energy E can be written as a postwaves F , using that dt ∂v ∂t Newtonian expansion in v/c, up to 3PN order for the energy (which only has integer orders) and 3.5PN order for the flux (which has both integer and half-integer orders). The fraction F (v)/E 0 (v) can be re-expanded in a number of different ways, which leads to different solutions due to the finite order of the expansions. As such, different PN approximations to the gravitational wave phase are obtained, which differ slightly (see for example Buonanno et al. (2009)). The general form is: Φ(v) =

n=7 Xh

φn + φ(l) n ln

n=0

 v i v n−5 c

c

.

(1.8)

Since the energy and flux are known up to 3.5PN order, the expansion goes up to n = 7, although in principle there are an infinite number of coefficients. These coefficients φn are functions of the binary parameters: two (l) masses and two spin vectors. The φn multiply an additional log term at some orders, for which the first (l) non-zero one is φ5 at 2.5PN. The amplitude of the wave can also be calculated, but higher order corrections to the amplitude are more difficult to measure than for the phase, which makes it of lesser interest for data analysis purposes. The approximation works well as long as v  c and the adiabatic approximations holds true, which fails for the late stages of the inspiral of a coalescing binary. For example, the peak velocity in the binary black hole merger GW150914 was v ∼ 0.6c. Furthermore, the PN expansion does not describe the merger and ringdown stages of the wave. However, models have been developed with input from numerical simulations, that are able to describe the full inspiral-merger-ringdown (IMR) stages of the gravitational waves. One of these is the IMRPhenomPv2 model, for which the phase is consists of three parts for the inspiral, intermediate (late inspiral) and merger-ringdown regions of the model (Khan et al., 2016): 
 φP N (M f ) + η1 σ0 + σ1 f + 34 σ2 f 4/3 + 53 σ3 f 5/3 + 12 σ4 f 2 for M f ≤ 0.018  
1 1 −3 for 0.018 < M f ≤ 0.5 fRD , (1.9) φ(f ) = η β0 + β1 f + β2 log(f ) − 3 β3 f

 f − α f  4 1 −1 −1 3/4 5 RD  α0 + α1 f − α2 f + α3 f + α4 tan for M f > 0.5 fRD η 3 fdamp where G = c = 1 for brevity, and fRD is the ringdown frequency of the final balck hole. φP N is a postNewtonian approximant as a function of GW frequency instead of velocity using Eq. (1.6). The other coefficients are determined with fits to numerical relativity waveforms, {σi } for the inspiral, {βi } for the intermediate and {αi } for the merger-ringdown regime. η = m1 m2 /M 2 is the symmetric mass ratio. This model, along with a different IMR model based on the Effective One Body formalism, will be further discussed in Chapter 3.

1.2

Detections of gravitational waves

In the first run (O1, from September 2015 till January 2016) of the Advanced LIGO (aLIGO) detectors, two detections of gravitational waves from merging binary black holes have been made with significance > 5.3σ, GW150914 and GW152126 (Abbott et al., 2016b,a). aLIGO consists of two laser interferometers, one in 7

Figure 1.2: Signals of binary black hole mergers in aLIGO’s O1 run. The plots show model reconstructions of the detected waveforms. The two detections are GW150914 with a (matched filter) SNR of 23.7 and GW151226 with an SNR of 13.0. LVT151012 is below the detection threshold with an SNR of 9.7. Figure taken from Abbott et al. (2016a). Hanford, Washington and one in Livingston, Louisiana. Having two detectors in different locations allows for a better distinction between an incoherent glitch, such as a seismic disturbance at one of the sites, and a coherent signal of a gravitational wave passing through both detectors within a light travel time. The interferometers are built of two perpendicular 4 km arms with mirrors at the end. A laser beam is split in two and sent up and down both arms, tuned such that the reflected beams are exactly out of phase. A passing gravitational wave will lengthen one arm while shortening the other (the exact effect depending on the strength of the polarisations of the wave and the angle of incidence on the detector), thus changing the length of the path traveled by the laser. This changes the phase of the laser beams such that they interfere differently, which is measured with a photo detector. Besides the two detections in O1, candidate signal LVT151012 was found to fall below the detection threshold with a significance of 1.7σ. (It was nevertheless investigated since it was observed coherently with both LIGO detectors and no specific disturbances in the data during the event were found.) All three signals match a model waveform of a merging binary black hole, for which parameters have been estimated (Abbott et al., 2016a). GW150914 was measured with a signal to noise ratio (SNR) of 23.7 and component masses +3.7 m1 = 36.2+5.2 −3.8 and m2 = 29.1−4.4 , given with 90% confidence intervals. GW151226 is a quieter event with +2.3 an SNR of 13.0, and component masses m1 = 14.2+8.3 −3.7 and m2 = 7.5−2.3 . The length of the measured signal depends on its frequency range compared to the sensitivity band of the detector, which is roughly from 30 Hz to ∼ 104 Hz. This is determined by the total mass of the system, which is inversely proportional to the frequency at merger. Hence, a much longer part of the inspiral is measured for GW151226, which has a lower mass and higher merger frequency, than GW150914. This difference can be seen in Fig. 1.2, which shows the model reconstructions of the three signals as given by Abbott et al. (2016a).

8

Chapter 2

Testing general relativity 2.1

The strong-field regime

In the century since Einstein’s publication of the theory of General Relativity (GR), GR has been successfully tested with many different observations (for a review, see Will (2014)). Since gravity is such a weak force, in almost any circumstance, only the linearised version of GR is necessary to describe physics. Therefore, the dynamical, strong-field regime of GR can not be tested with these observations. This holds even for the most relativistic binary neutron star that has been observed electromagnetically, the binary pulsar PSR J07373039. Gravitational wave observations of merging compact binaries, such as GW150914, do allow the study of the dynamical, strong-field regime. Compare the orbital velocity of PSR J0737-3039 of v ∼ 2 × 10−3 c to that of GW150916, which is v ∼ 0.6c at its peak. The gravitational field strengths can be expressed with the dimensionless quantity GM/(c2 R), where M is the binary’s total mass and R its separation, being 4.4 × 10−6 for PSR J0737-3039 and ∼ 0.1 for GW150914. If reality is governed by some alternative theory to GR that manifests itself in the strong-field regime, this should leave a specific mark on gravitational waves emitted by binary coalescences. It would be relatively straightforward to devise a test that discriminates between said alternative theory and GR. However, a new test would be needed for each alternative theory, and even then possible unknown alternatives have not been accounted for. It is more desirable to devise a generic test, that can detect any possible deviation from GR. The TIGER method (Li et al., 2012) is an implementation of such a generic test. It uses Bayesian inference to calculate the ratio between the evidence for GR and the evidence for “not GR”, called the Bayes factor (the probability ratio of the two hypotheses with the ratio of prior probabilities taken out). It is a computationally expensive calculation, but part of its infrastructure can be used on its own to constrain deviations from GR. This part, which will be referred to as the parameterised test of GR, has already been performed using the data from GW150914 and GW151226 (Abbott et al., 2016d,a). Results of these tests can be found in section 2.2.2.

2.2

The parameterised test of GR

In the TIGER method, the hypotheses “gravity follows GR” and “gravity deviates from GR” are formalised using the coefficients of a waveform. For an inspiral gravitational wave signal, a post-Newtonian expansion could be used, which has convenient coefficients for each order in v/c (see Eq. (1.8)). For the inspiral-mergerringdown (IMR) waveforms from coalescing binaries, the IMRPhenomPv2 model is used (see Eq. (1.9) and section 3.1.2), which has besides the PN coefficients {φi }, additional coefficients {σi }, {βi } and {αi }. Say all the coefficients of the chosen model are the set {φn }. Then, the GR hypothesis in the TIGER method is “All of the coefficients φn are as predicted by GR”. Every other possibility is captured by the non-GR hypothesis, “One or more of the coefficients φn are different from the value as predicted by GR”.

9

The phrase “one or more of the coefficients” can be described equivalently as “any of the non-empty subsets of {φn }” (the empty subset corresponds to the GR hypothesis). Each subset S can be associated to a sub-hypothesis HS , “the coefficients in the set S deviate from the GR value, and other coefficients do not”. For example, in a waveform model with two coefficients {φ1 , φ2 }, the non-emtpy subsets are {{φ1 }, {φ2 }, {φ1 , φ2 }}, corresponding to the sub-hypotheses H1 , H2 , and H1,2 . In order to parameterise the deviations from GR, each coefficients φn is multiplied by a factor (1 + δ φˆn ) in the waveform model, such that all δ φˆn = 0 represents the GR waveform. The set {δ φˆn } is called the deviation parameters. In the full TIGER calculation, the evidence for each of the sub-hypotheses is calculated separately, after which they are combined to form the evidence for the non-GR hypothesis. (The non-GR hypothesis is the logical OR of the sub-hypotheses. Since they are logically disjoint, the evidences can be averaged over, see also Van Den Broeck (2014).) However, before calculating the combined Bayes factor, valuable results can be obtained from the evidence calculations on the sub-hypotheses, especially those corresponding to a single deviation parameter. The evidence for the sub-hypothesis Hk , corresponding to δ φˆk 6= 0, is formally given by the probability of measuring the data d, given that Hk and any background information I are true: Z ~ Hk , I) p(θ|H ~ k , I). P (d|Hk , I) = dθ~ p(d|θ, (2.1) The integral over θ~ represents a marginalisation over the parameter space of the model, which includes all the parameters from GR, such as the masses and spins of the binary, as well as the deviation parameter δ φˆk . ~ k , I) is the prior probability distribution of the parameters, and p(d|θ, ~ Hk , I) The probability density p(θ|H is the likelihood function. The likelihood is calculated from the data, by matching it with a modelled signal. This integral is evaluated using the nested sampling algorithm (Skilling, 2004), as explained in the following section 2.2.1. Using the calculation of the evidence integral with nested sampling, it is trivial to extract the posterior probability distribution on any parameter as well. For the posterior of the deviation parameter δ φˆk , with Bayes’ theorem, we have: p(d|δ φˆk , Hk , I)p(δ φˆk |Hk , I) . (2.2) p(δ φˆk |d, Hk , I) = P (d|Hk , I) The denominator is given by Eq (2.1), and the numerator can be found by integrating the same equation over only the parameters found in GR, leaving out the integration over δ φˆk : Z ˆ ˆ ~ Hk , I) p(θ|H ~ k , I). p(d|δ φk , Hk , I)p(δ φk |Hk , I) = dθ~GR p(d|θ, (2.3) If the data d is determined by GR, the distribution (2.2) should be peaked around δ φˆk = 0, or possibly with a small offset from zero due to noise in the data. Given the width of the distribution, one can constrain the value of δ φˆk to a certain interval (around zero in the case of GR), and thus exclude any values outside the support of the distribution. This then, is the parameterised test of GR; constraining each deviation coefficient by its posterior probability distribution. The data d represents a signal from a single binary coalescence event, but data from multiple events can easily be combined. Given data from two distinct events, dA and dB , the combined posterior probability distrubtion of the deviation parameter δ φˆi can found using Bayes’ theorem. It is also assumed that the data are independent, such that p(dA |dB ) = p(dA ) and vice versa, which yields: p(δ φˆk |dA , dB , I) =

p(δ φˆk |dA , I)p(δ φˆk |dB , I) . p(δ φˆk , I)

(2.4)

Equivalently, any number of posterior distributions can be combined by multiplying them, and dividing by the prior of δ φˆk for any event after the first. 10

2.2.1

The nested sampling algorithm

The nested sampling algorithm can be used to calculate the integrated evidence from Eq. (2.1) by producing ~ Hk , I) and the prior p(θ|H ~ k , I) over parameter space. Conveniently, these samples of the liklihood p(d|θ, ~ for samples can also be used to calculate the posterior probability distribution of any of the parameters in θ, example those for the parameterised test of GR as in Eq. (2.2), for which the total evidence is a normalisation factor. In the following, the algorithm will be explained by evaluating the evidence first and from there the posterior distribution, although the latter is of greater interest for the parameterised test; it is the same algorithm either way. In evaluating the evidence integral, first, the likelihood and prior need to be quantified. The total prior is simply a multiplication of a chosen prior for each of the model parameters. Since there is no a priori reason to prefer any parameter values, the priors are mostly uniform distributions on an astrophysically or computationally sensible interval (Abbott et al., 2016c). The two angles determining the sky position (right ascension and declination), and the inclination of the source are naturally limited on different intervals ([0, π], [−π/2, π/2] and [0, π/2] respectively). The distance is the only non-flat prior, since it is chosen such that the distribution of sources is homogeneous over a volume. A maximum distance can be chosen such that even the brightest sources would not be detectable from further away, and so an interval of [1, 2000] Mpc is used. The spin magnitudes are naturally limited on [0, 1], and the spin orientations on the sphere; flat priors are chosen for both. For the initial searches, the component masses are limited on [1, 100] M , but for later analyses this interval can be tightened to decrease computational cost, depending on the estimated parameters. For GW150914, a mass prior of [10, 80] M is used, and for GW151226 this is [3.022, 54.398] M . Lastly, the prior for each deviation parameter is chosen wide enough so that it has no effect on the measurement. ~ predicted by Hk . The likelihood function expresses how well the data fit the model waveform hk (θ) Assuming the data d consist of a signal and stationary, gaussian noise, the likelihood is defined as follows (Veitch and Vecchio, 2010; Van Den Broeck, 2014): ~

~

~ Hk , I) = N e−(d−hk (θ)|d−hk (θ))/2 . p(d|θ,

(2.5)

Here, N is some normalisation factor. This holds for data from a single detector, but data from multiple detectors can be combined by multiplying the likelihood functions. The inner product that is used is given by: Z ∞ ˜∗ h (f ) g˜(f ) df, (2.6) (h|g) = 4Re Sn (f ) 0 ˜ and g˜ are the Fourier transformed functions, and h ˜ ∗ represents the complex conjugate. Sn (f ) is where h the noise power spectral density, i.e. the power of the noise per frequency bin df , as a function of frequency. Its form depends on the characteristics of the detector, and can be measured in the absence of a signal. In practice, either an analytical estimate or a fit based on noise measurements is used. A plot of a typical noise curve for the Advanced LIGO detector is given in figure 3.2.1. Since the noise gets loud for either low or high frequencies, the integral in Eq. (2.6) can be cut off on both sides. Furthermore, since the data are not continuous but a discrete set of points, the integral is replaced by a summation, and the continuous Fourier transform is replaced by the discrete version. Having quantified the evidence integral (2.1), it is not trivial to calculate over a highly dimensional parameter space as the one for gravitational wave models. Veitch and Vecchio (2010) have described a method to make this computationally feasible using the nested sampling algorithm (Skilling, 2004). As the integral can not be analytically computed, it needs to be approximated by a sum over N points in parameter space: Z N X ~ Hk , I) p(θ|H ~ k , I) ≈ dθ~ p(d|θ, Li wi , (2.7) i=1

11

where Li is the likelihood at the point θ~i (as in Eq. (2.5)), and wi is the fraction of the prior associated to it. A simple way of distributing N values is to lay a regular grid over parameter space, since then each weight is given by wi = 1/N (for a flat prior). However, only a small fraction of the points contribute the most to the total sum, since the likelihood is only high in a small section of parameter space, namely that where the model waveform is close to the signal. Thus, it would require many points to get an accurate approximation of the integral (see (Veitch and Vecchio, 2010) for a quantification). In the nested sampling algorithm, instead, only relatively few points are used at any time to sample the likelihood, and the results are used to evolve the collection of points to regions of higher likelihood iteratively. At each step, the contribution to the integral of the lowest likelihood point is calculated, after which it is replaced by randomly drawing a new point, on the condition that is has a higher likelihood than the original point. A remaining difficulty is assigning the weights wi to the sample points. Imagine arranging the points by increasing likelihood, and each point laying on an equal likelihood surface enclosing a volume in parameter space. The total volume of parameter space is assigned a prior probability of one, and so each smaller volume represents a fraction of the prior. Moving along the collection of points towards higher likelihood values, the enclosed fraction of the prior decreases with each step. The weight wi assigned to a point θ~i , then, is the difference between its enclosed fraction and that of the previous point. These weights can not be determined exactly, but a probability distribution is found for the ratio between consecutive prior fractions, such that the average ratio can be calculated. This number is used to approximate the consecutive fractions, and thereby the weights wi , also using that the outermost possible sample point would enclose the total prior of one. (For details, see Veitch and Vecchio (2010).) The algorithm is terminated on the basis of the highest likelihood point that has been found so far. An estimate of the remaining evidence is obtained by assuming that all points are at this highest likelihood value. If this is less than a small fraction (e−5 ) of the total evidence accumulated so far, the iterations are stopped and the estimated remaining evidence is added to the total. This describes the complete outline of the algorithm, although it can be run multiple times (simultaniously) to obtain a more accurate result. In that case, each point θ~i is saved along with its likelihood value Li , after which the combined set of points is rearranged and new weights wi are assigned. As mentioned before, the calculation of the total evidence can also be used to find the posterior probability density of the deviation parameter(s). This is done by re-using the samples Li wi , and the total evidence as a normalisation factor. Given equations (2.2) and (2.3), the postetior distribution of the deviation parameter δ φˆk is given by: R ~ Hk , I) p(θ|H ~ k , I) dθ~GR p(d|θ, ˆ (2.8) p(δ φk |d, Hk , I) = P (d|Hk , I) As in equation (2.7), the integral can by approximated by a sum over the posterior samples Li wi , only here the samples are first divided over bins in δ φˆk before they are summed. However, the weights need to be corrected for the increased density of samples towards higher likelihood. For sample Li wi , where the likelihood contour at Li encloses a fraction Xi of the prior, the sample density is boosted by a factor 1/Xi . To compensate, corrected weights wi Xi are used. Note that, outside of the parameterised test of GR, other model parameters are estimated in the same way as a deviation parameter.

2.2.2

Results for GW150914 and GW151226

For the parameterised test of GR on data from GW150914 and GW151226, the waveform model IMRPhenomPv2 (see sections 3.1.2 and 3.1.2) was used. This phenomenological model is characterised by a set of coefficients for the phase and amplitude of the wave, both divided into three parts for the inspiral, intermediate and merger-ringdown regimes of the model. Only the phase coefficients are used in the parameterised test, since it is more accurately measured than the wave’s amplitude. Some coefficients are excluded because they have the same effect as a constant shift in the wave’s initial phase φ0 or starting time t0 , and therefore can not be measured. Furthermore, the set of phenomenological coefficients that provide corrections to the post-Newtonian description of the inspiral phase are excluded as well. They are only determined with high uncertainties in the calibration of the model. As their GR predicted value is not very precise, it 12

would be difficult to find a significant deviation in any signal with these coefficients. For each remaining parameter in the testing set, a deviation coefficient is built in the model as explained in section 2.2. They are {δ χ ˆ0 , δ χ ˆ1 , δ χ ˆ2 , δ χ ˆ3 , δ χ ˆ4 , δ χ ˆ5l , δ χ ˆ6 , δ χ ˆ6l , δ χ ˆ7 } for the inspiral regime (see the inspiral parameters {φn } in appendix A in Khan et al. (2016), only note that the logarithmic parts of φ5 and φ6 are treated as separate coefficients indicated with a subscript l), {δ βˆ2 , δ βˆ3 } for the intermediate, and {δ α ˆ2 , δα ˆ3 , δα ˆ 4 } for the merger-ringdown regime (see Table V in Khan et al. (2016)). The results of the parameterised test on GW150914 and GW151226 published in Abbott et al. (2016d,a), are given in Fig. 2.1. The combined posteriors, as in Eq. (2.4), are found in the bottom panel of the same figure. The posteriors of the deviation coefficients on the post-Newtonian parameters of GW150914 are not well centered around zero. This may be due to the short length of the inspiral part of the waveform that was measured, in combination with effects of the detector noise. For GW151226, for which a much longer part of the inspiral was in band for aLIGO, the posterior are much more centered on the GR value. An investigation on the effects of noise is discussed in section 4.1.

13

Figure 2.1: Results of the parameterised test of GR on GW150914 and GW151226, as published in Abbott et al. (2016a). The posteriors are given as violin plots, with bars indicating the 90% confidence interval. The top two rows are results from the individual events, while the bottom row shows the combined posteriors. Coefficients from the inspiral regime of the model IMRPhenomPv2 are indicated by their post-Newtonian order. The last panel contains the tested coefficients from the intermediate (βn ) and merger-ringdown (αn ) regimes.

14

Chapter 3

Waveforms In the analysis of a gravitational wave (GW) signal from a binary coalescence, many different model waveforms need to be generated. For example, to calculate posterior distributions for the binary’s parameters, template waveforms are needed across the possible parameter space to be compared to the measured waveform. Exact solutions to the Einstein equations only exist for some very specific mass-energy configurations, such as a single black hole or a homogeneous and isotropic universe. For other cases they can be solved numerically, however, numerical relativity (NR) does not suffice for these GW analyses. It takes NR codes roughly a month to generate a single waveform. Moreover, in the parameterised test, a non-GR parameter is included in the search. Template non-GR waveforms can not be produced using NR, since it simulates GR spacetime. Gravitational waves from coalescing binaries will have to be modelled with approximations. Two of these have already been discussed in Section 1.1, the first being the post-Newtonian (PN) expansion in powers of v/c. This is a very good approximation as long as the orbital velocity is not very high, which is true during the early inspiral of the binary, but not when the objects get closer to each other and merge. For signals from binary neutron stars, waveforms using the PN approximation are sufficient, because the merger happens at such a high frequency that it is not well measured by the detector anyway. On the other hand, for binaries with one or two black holes, the late inspiral and the merger can be in band of the detector. Additionally, the dynamics are complicated by the spins of the black hole(s) and the presence of a typical ringdown signal of the new black hole that was created during merger. As these more complicated parts of the signal contain a lot of interesting information - they are created in a stronger, more dynamical gravitational field - a waveform is required that can model this complexity. Different approximation models exist, such as the IMRPhenomPv2 model that was briefly discussed in Section 1.1. Another important method used the Effective One Body (EOB) formalism, and both are explained in the first section 3.1 of this chapter. In both methods, the model is calibrated against a set of NR waveforms. It can be easily verified that the models are within reasonable error from both the NR waveforms used for calibration and others within the calibration parameter range. However, a different test is needed to assess the accuracy of the models at any point in parameter space, where no NR waveform exists. Section 3.2 contains the methods and results of a test that has been done based on comparing different approximant models to each other.

3.1

Waveform approximations

Both methods of approximating GR waveforms discussed in this section take the post-Newtonian expansion of GR as a starting point, and manage its accuracy by calibrating the waveform to NR results. As the PN model can not (accurately) describe the late inspiral, merger and ringdown of a GR waveform, it has to be extended in some way. An additional difficulty in modelling these waveforms are possible precession effects ~ · L, ~ where S ~ is one of the of the black hole spins. To leading order, these are due to spin-orbit coupling S

15

~ the orbital angular momentum of the binary. For spins aligned with L, ~ there is no black hole spins and L precession.

3.1.1

Effective One Body

The EOB formalism The EOB formalism is a method that maps the dynamics of a two-body system (for example, a black hole binary), to the dynamics of one (extended) body and a test particle (for a more detailed explanation, on which the following is based, see Damour (2013), or Buonanno et al. (2009)). As a starting point, it uses the Hamiltonian of a two-body system in GR, that has been calculated using a post-Newtonian expansion up to 3PN order. The first, non-GR term (0PN) describes a ’test particle’ of mass µ that moves in the gravitational field of an ’external mass’ M . Here, M is the total mass of the two bodies and µ the reduced mass µ = m1 m2 /M . The idea behind EOB is to extend this description to GR by finding an effective geometry for the test particle to move in, such that its dynamics are equivalent to the actual system. An ansatz is made for the effective geometry (Buonanno et al., 2009): ds2ef f = −A(r)dt2 +

D(r) 2 dr + r2 (dθ2 + sin(θ)2 dφ2 ) A(r)

(3.1)

Where A(r) and D(r) are written as post-Newtonian expansions: Ak (r) =

i=k+1 X i=0

Dk (r) =

ai (η) ri

i=k X di (η) i=0

ri

(3.2a)

(3.2b)

ai (η) and di (η) are all functions of the symmetric mass ratio η = µ/M that need to be determined by the mapping between the actual and effective dynamics. To what order k this can be done, depends on the PN order of the dynamics of the original problem; for 3PN, k = 4. The geometry should reduce to the Schwarzschil geometry for η → 0, as that describes the case of an actual test particle of zero mass around a single mass M . This requirement sets some of the coefficients to a fixed value. The mapping between the two descriptions is inspired by quantum mechanics: the energy levels are computed as though they were quantised (only here the quantum numbers n and l are very large). The energy levels for the 3PN dynamics can be calculated from the Hamiltionian. Similarly, the energy levels of the effective description can be calculated from the geometry, with quantum numbers nef f and lef f . A unique mapping between the energy levels of the actual and the effective dynamics can be found by identifying n = nef f , l = lef f . This solution then completes the description of the effective dynamics. Having a description of the dynamics of the EOB system, the gravitational radiation needs to be introduced. This is done by adding a radiation reaction force in the equation of motion of the momentum in the azimuthal direction, assuming that the binary is circularised. After resumming a number of quantities, explicit expressions for the radiation reaction and the emitted waveform are obtained. The waveform produced by the EOB method encompasses only the inspiral part of the binary coalescence. In order to complete it, a superposition of quasi normal modes (QNM) is pasted after it, which represents the ringdown of the Kerr black hole that was formed. The cut is made at the peak of the orbital frequency, which is identified as the time of effective merger tm . The coefficients of the QNM’s are adjusted so that the two parts of the waveform (and its time derivative) match each other at a number of points chosen around tm . The complete waveform now consists of the EOB waveform up until tm , and the matched ringdown waveform after tm . (Damour, 2013; Buonanno et al., 2009)

16

EOBNR waveforms The complete EOB waveform is constructed off of a post-Newtonian expansion of the dynamics of the binary. It is therefore not surprising that these waveforms are not quite accurate compared to numerical relativity (NR) waveforms. In order to fix this, input from the NR waveforms is used. Higher order PN terms are added in the dynamics, although the coefficients are unknown. The complete EOB waveform is calculated with the undetermined coefficients. Additionally, unknown non-quasi circular parameters are added to the waveform, to account for a departure from an adiabatic quasi-circular inspiral as described by the PN expansions. Then, the unknowns are determined by fitting the EOB waveform to NR data. The result is calibrated EOBNR waveforms, that are much more accurate than the original EOB waveforms. (Damour, 2013) The EOBNR method has been extended to include spin, and the newest waveform model including spins aligned with the orbital angular momentum (i.e. non-precessing spins) is SEOBNRv2 (Taracchini et al., 2014). It has been calibrated using 27 of a total of 38 NR waveforms listed in Table 3.1.1. In the calibration set, mass ratios range from 1.0 to 8.0. Spins range from -0.95 to 0.98, but only for q = 1, whereas the largest spin value for unequal mass binaries is 0.5. nr. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

q 1.0 1.5 2.0 3.0 4.0 5.0 6.0 8.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

η 0.25 0.24 0.22 0.19 0.16 0.14 0.12 0.099 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25

χ1 0 0 0 0 0 0 0 0 0.98 0.97 0.95 -0.95 0.9 -0.9 0.85 0.8 -0.8 0.6 -0.6

χ2 0 0 0 0 0 0 0 0 0.98 0.97 0.95 -0.95 0.9 -0.9 0.85 0.8 -0.8 0.6 -0.6

nr. 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

q 1.0 1.0 1.0 1.0 3.0 3.0 1.0 1.0 1.5 1.5 3.0 3.0 5.0 5.0 8.0 8.0 1.5 1.5 2.0

η 0.25 0.25 0.25 0.25 0.19 0.19 0.25 0.25 0.24 0.24 0.19 0.19 0.14 0.14 0.099 0.099 0.24 0.24 0.22

χ1 0.44 -0.44 0.2 -0.2 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.6

χ2 0.44 -0.44 0.2 -0.2 0.5 -0.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.5 0.5 0.0

Table 3.1: NR waveforms used in the calibration of the SEOBNRv2model (Taracchini et al., 2014). 1-8 are non-spinning waveforms, and 9-25 have equal spins (χ1 = χ2 ). Furthermore, a waveform model with precessing spins has been implemented under the name SEOBNRv3 (Pan et al., 2014). Its strategy is to compute the precession effects separately form the rest of the waveform, ~ In this frame, the inspiral by constructing a frame that precesses along with the orbital angular momentum L. ~ EOB waveform is produced using the spin components aligned with L only. Then, it is transformed to the frame aligned with the spin of the final black hole, in which the ringdown waveform (the superposition of QNMs) is added and matched around the merger time. Finally, another transformation yields the waveform in the binary’s original orientation. The non-precessing inspiral waveform is produced using the SEOBNRv2 model, and no further (precessing) calibration NR waveforms are used in SEOBNRv3.

17

Reduced order modelling Although an EOBNR waveform is much faster generated than a full NR waveform, it is not a closed-form analytical formula as it involves numerically solving differential equations. For example, it takes SEOBNRv3 about 2.4 seconds to make a waveform for a binary black hole with total mass 60M (equal masses and zero spins). This makes is still too slow to be of any use when calculating posteriors with the parameterised test. It is still useful for creating mock data by injecting a model waveform in detector noise, since then only one waveform needs to be calculated. A method exists to turn a slower model, such as SEOBNRv2, into a faster one. Reduced Order Modelling (ROM) (P¨ urrer, 2014) uses a basis of many different pre-calculated waveforms across parameter space and interpolates for any given point between them. The basis is chosen such that it captures all the most important features of the model. Different versions of a ROM model for SEOBNRv2 exist, of which SEOBNRv2ROMDoubleSpin is the most recent version that can take any two values for the aligned spins of the two compact objects. To compare with SEOBNRv3, this model produces a waveform with M = 60M in about 0.036 seconds. It has been shown that the error of the ROM model relative to the original model is small compared to the error in the original model to NR waveforms (P¨ urrer, 2014), hence it is just as accurate to use the ROM model. Unfortunately, no ROM model exists for SEOBNRv3. Since this approximant uses precessing spins, there are four additional parameters compared to SEOBNRv2 (the x-components and y-components of both spins, although their in-plane angle is usually not very relevant). A ROM model would therefore require a basis of calculated waveforms over a parameter space with four additional dimensions, which is a non-trivial upgrade.

3.1.2

Phenomenological models

A different approach to making a fast waveform approximant is building a phenomenological model, for which the procedure (Husa et al., 2016; Khan et al., 2016) is outlined here. First, example waveforms are made by combining an approximate waveform for the inspiral - for example from a PN expansion - and numerical relativity (NR) data for the late inspiral, merger and ringdown. These hybrid waveforms are constructed in the frequency domain, as is the eventual phenomenological model, which is convenient for use in most analysis methods. For this purpose, the NR data is converted with a discrete Fourier transform. The parameters of the inspiral waveform are adjusted so that it matches the NR waveform well, over a chosen frequency window in which both are reliable approximations. Then, these hybrid waveforms are used to calibrate a phenomenological ansatz. The ansatz is a closedform analytical expression with, before the calibration, unknown phenomenological parameters. These are expressed as functions of the masses (m1 , m2 ) and spins (χ1 , χ2 ) of the binary. With these functions, any point in parameter space can be used to obtain a waveform. Generally, the approximation is only valid in the region of parameter space that was covered by the hybrid waveforms used for calibration. PhenomD IMRPhenomD (Husa et al., 2016; Khan et al., 2016) is the latest in a series of phenomenological models with non-precessing spins. It uses hybrids of uncalibrated SEOBNRv2 and 19 different NR waveforms to fit the model to. The range of NR waveforms includes mass ratios up to 16, and non-precessing spins from -0.95 to 0.98 for equal mass binaries, and roughly in between ±0.8 for non-equal masses (see Table 3.1.2 or Table 1 in Khan et al. (2016)). Configuration with spin magnitudes greater than 0.85 and non-equal masses for example, are not supported by this calibration range. The ansatz is split into two different models for the wave’s amplitude and phase, which are in turn split into three parts for the inspiral, intermediate and merger-ringdown regions of the model (Husa et al., 2016;

18

Khan et al., 2016). The complete IMR phase is given by: 
3 3 1 4/3 5/3  + 12 σ4 f 2 φP N (M f ) + η σ0 + σ1 f + 4 σ2 f
+ 5 σ3 f  1 1 −3 φ(f ) = η β0 + β1 f + β2 log(f ) − 3 β3 f

   1 α0 + α1 f − α2 f −1 + 4 α3 f 3/4 + α4 tan−1 f − α5 fRD η 3 fdamp

for M f ≤ 0.018 for 0.018 < M f ≤ 0.5fRD , for M f > 0.5fRD

(3.3)

where η = m1 m2 /M 2 is the symmetric mass ratio, and G = c = 1. For the inspiral region, both the phase and amplitude are based on the TaylorF2 model, a known post-Newtonian expansion in the frequency domain (see, for example, Buonanno et al. (2009)), for which the phase is denoted φP N . Some higher order terms are included with extra phenomenological parameters {σi } as prefactors. At M f = 0.018, the example waveforms consist of purely NR data, at which point the intermediate regime starts. For this region as well as the merger-ringdown, purely phenomenological guesses are used based on the evolution of the amplitude and phase, with coefficients {βi } for the intermediate and {αi } for the merger ringdown regime. Some of these, namely β0 , β1 , α0 and α1 , are used solely to ensure the continuity of the phase and its derivative. The boundary between the two regions is at M f = 0.5fRD , where fRD is the ringdown frequency of the final black hole which is calculated from its mass and spin. fdamp , used in modelling the ringdown, is the damping frequency of the final black hole, which can also be calculated from the mass and spin parameters. The inspiral coefficients {σi } are expressed as polynomials in the physical parameters m1 , m2 , χ1 and χ2 . Since this is a non-precessing model, χ1 and χ2 are considered as the spin components parallel to the orbital angular momentum. In the fit of the ansatz to the hybrid waveforms, the coefficients of the polynomials are determined. (The exact expressions can be found in appendix B of Khan et al. (2016).) This allows for the coefficients, and thus the complete model, to be constructed with any choice of physical parameters. The intermediate and merger-ringdown coefficients are also written as polynomials, but only using the physical parameters η (the symmetric mass ratio) and χP N , a combination of the spins of the two objects. χP N is chosen as it determines the leading-order spin effect in the phase of the gravitational wave. Using it instead of the two spins is a good approximation for most binaries, but may not hold for high mass ratios or high spins. Its value is given by (see also Eq. (2) and (3) in Khan et al. (2016)): m1 χ1 + m2 χ2 , M 38η = χe − (χ1 + χ2 ). 113

χe = χP N

(3.4) (3.5)

Effective precession The IMRPhenomD model is only applicable to binaries with non-precessing spins, i.e. spins aligned with ~ A full spin phenomenological model would require using four additional the orbital angular momentum L. physical parameters (two additional spin components for each object) in the ansatz, and many more NR waveforms to cover the extended parameter space. However, a method has been developed by Hannam et al. (2014) to turn a non-precessing into a precessing waveform model; no additional calibration with precessing waveforms is used. The resulting model for precessing spin configuration is called IMRPhenomPv2. In the method by Hannam et al. (2014), the complete inspiral-merger-ringdown waveform is calculated first using only the spin components aligned with the orbital angular momentum, a valid approximation if the in-plane spin components do not have a significant effect on the inspiral. This is true as long as the direction of the total angular momentum J~ does not change (transitional precession), which is expected to ~ around J~ is modelled using the happen only in rare cases (Schmidt et al., 2015). Then, the precession of L in-plane spin components, and used to find a rotation that transforms the non-precessing waveform into a precessing one. In general, this rotation is defined by two angles that depend on all four in-plane spin components and the binary’s mass ratio. They can be calculated using leading-order (or higher) PN equations for the precession. For a generic spin configuration, there are no analytic solutions to these equations, but for the special case 19

nr. A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19

q 1.0 1.0 1.0 1.0 1.0 4.0 4.0 4.0 4.0 4.0 8.0 8.0 8.0 8.0 8.0 18.0 18.0 18.0 18.0

η 0.25 0.25 0.25 0.25 0.25 0.16 0.16 0.16 0.16 0.16 0.099 0.099 0.099 0.099 0.099 0.05 0.05 0.05 0.05

χ1 -0.95 -0.6 0.0 0.6 0.98 -0.75 -0.5 0.0 0.5 0.75 -0.85 -0.5 0.0 0.5 0.85 -0.8 -0.4 0.0 0.4

χ2 -0.95 -0.6 0.0 0.6 0.98 -0.75 -0.5 0.0 0.5 0.75 -0.85 0.0 0.0 0.0 0.85 0.0 0.0 0.0 0.0

χ ˆ -0.95 -0.6 0.0 0.6 0.98 -0.75 -0.5 0.0 0.5 0.75 -0.85 -0.458 0.0 0.458 0.85 -0.77 -0.385 0.0 0.385

Table 3.2: NR waveforms used in the calibration of the PhenomD model. Adapted from Khan et al. (2016), using the same identification numbers. Given are the mass ratio q, the symmetric mass ratio η, both dimensionless spins χ1 and χ2 , and χ, ˆ which is χP N normalised between -1 and 1 (see Eq. (4) in Khan et al. (2016)). where one of the black holes has zero in-plane spin, there are. In the model by Hannam et al. (2014), the four spin components are reduced to one effective precession spin parameter χp . A single spin configuration is obtained by assigning χp as the spin of the larger black hole, which serves as an approximation of the full configuration. The motivation for χp is presented here, following Schmidt et al. (2015). It starts by considering the leading-order precession effect, namely spin-orbit coupling at 1.5PN order:   ~˙ = L A1 S ~1⊥ + A2 S ~2⊥ × L. ˆ L r3

(3.6)

3 Here, r is the separation and A1 = 2+ 2q , A2 = 2+ 3q 2 . (Although Schmidt et al. (2015) define q ≡ m2 /m1 > 0, the equations are adapted to the convention of q ≡ m1 /m2 > 0 used throughout this work.) ~ their relative angle changes continuously. As the spins both precess at, in general, different rates around L, Therefore, the term in square brackets in Eq. (3.6) varies between its minimum when the two spins points in opposite directions and its maximum when the two spins are aligned. An approximation is made for the average of the spin-term, given by the average of the minimum and maximum values:

Sp ≡

1 (A1 S1⊥ + A2 S2⊥ + |A1 S1⊥ − A2 S2⊥ |) = max(A1 S1⊥ , A2 S2⊥ ). 2

(3.7)

The spin Sp can either be determined by the spin of the larger or the smaller black hole. Usually, S1⊥ dominates the precession, so in this approximation all spin is assigned to the larger black hole. This then defines the effective spin parameter: Sp . (3.8) χp ≡ A1 m21 In some cases, the representation of two in-plane spin components with χp is not a good approximation. Two assumptions were made in the motivation of χp that can both be invalid. The first was that both 20

~ such that the weighted sum of the two in-plane spin components spins precess at different rates around L, could be approximated by Sp as in Eq. (3.7). The rate of precession depends on the mass of the black hole, thus the rates are (almost) the same for (almost) equal mass systems. In such a case, the relative angle ~1⊥ + A2 S ~2⊥ , and therefore between the two in-plane spins will be constant. This determines the size of A1 S the strength of the spin-orbit coupling. For example, for equal masses and an in-plane angle φ = 0, the spin components add directly and the precession effect will be underestimated by χp . If on the other hand φ = π, the spin components cancel out and there is no spin-orbit coupling at all. Schmidt et al. (2015) found that this becomes a problem for mass ratios q . 1.2, although the effect is comparable to that of neglecting the higher order spin-spin coupling. The second assumption was that the larger black hole will dominate the spin-orbit coupling, which is why χp is assigned as its spin. However, this is not the case if A2 S2⊥ > A1 S1⊥ . Using χi ≡ Si /m2i , this condition can be rewritten as: A2 χ2⊥ . (3.9) χ1⊥ < A1 q 2 For q ' 1.0, this is simply when the in-plane spin of the smaller black hole is larger than that of the bigger black hole. The higher the mass ratio, the greater the difference between the spins must be for the condition to be met. (For large q, the factor A2 /(A1 q 2 ) is proportional to 1/q.) Schmidt et al. (2015) note ~ and J~ - is affected by this problem, only a that although one of the precession angles - the one between L small percentage of the resulting waveforms deviate significantly from waveforms constructed with the full spin configuration. The problems occurs mostly for configurations where the in-plane spin component of the larger black hole is small, χ1 ≤ 0.08.

3.2

Overlaps

In order for a waveform model to do its job, it needs to be both fast and approximate the actual GR solution accurately. The parameterised test of GR requires the calculation of so many instances of the waveform, that only IMRPhenomPv2 and SEOBNRv2ROM meet the speed requirement. As the latter admits no precessing spins, IMRPhenomPv2 is the usual approximant of choice for these kinds of searches. It is therefore important that its accuracy is verified. The most direct way of checking a GR approximation is comparing the waveform to NR results. But NR waveforms only exist for a select set of binary parameters. A test that allows any choice of parameters is comparing two waveform models to each other. If they both accurately represent the GR solution at that point, they should be very similar. Such a test is discussed in this section, in which the overlap between IMRPhenomPv2 and SEOBNRv2ROM or SEOBNRv3 is examined. Note that a similar test for IMRPhenomD and SEOBNRv2 has already been done by Khan et al. (2016), for a number of different masses and mass ratios with equal spin configurations (see also the discussion of the results 3.2.4). Here, the test will be extended to non-equal spins and precessing spins for the waveforms IMRPhenomPv2 and SEOBNRv3.

3.2.1

Faithfulness

In order to compare two waveforms, a quantitative measure for their similarity is required. As is used in the calculation of the evidence integral (see section 2.2.1 for a definition of the symbols), the inner product between to waveforms is defined as: Z ∞ ˜∗ h (f ) g˜(f ) (h|g) = 4Re df. (3.10) Sn (f ) 0 Sn (f ) is the noise power spectral density (PSD), i.e. the power of the noise per frequency bin df , as a function of frequency. Its form depends on the characteristics of the detector, and can be either an analytical estimate,

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or a fit based on measured noise. In this work, an estimate of the noise curve for the early stages of Advanced LIGO (Abbott et al., 2013) is interpolated and the integral is cut off from 20 Hz to the lowest maximum frequency of the two waveforms that are used. A plot of the noise PSD is given in Fig. 3.2.1. For the product to be of practical use as a similarity measure, the waveforms are normalised such that (h|h) = 1, ensuring that the product of two different waveforms always lies in the range of minus one to one, with negative values for anti-correlated waveforms. Each waveform can be given an arbitrary start time t0 and an arbitrary phase at this time, φ0 . It makes sense to maximise over these two parameters when matching two waveforms (either a model and a signal or two models). Applying this to Eq. (3.10) yields the faithfulness: (h(λ)|g(λ)) p . (3.11) F(h, g) = max p t0 ,φ0 (h(λ)|h(λ) (g(λ)|g(λ) Here, the faithfulness F is evaluated for the physical parameters λ. When matching a waveform model to a signal with unknown parameters λ0 , the match is maximised over λ as well. The overlap measure that is obtained in this way is called the effectualness: (h(λ0 )|g(λ)) p . E(h, g) = max p t0 ,φ0 ,λ (h(λ0 )|h(λ0 ) (g(λ)|g(λ)

(3.12)

A waveform model that is effectual does not have to be faithful as well; it could be that the maximum match is obtained for model parameters λ that deviate significantly from the signal parameters λ0 . It would be sufficient for the detection of a signal, but not suitable for parameter estimation. As we are concerned with the accuracy of waveform models for given parameters, the measure of choice is the faithfulness.

Figure 3.1: Predicted noise curve for the early stages of the Advanced LIGO detectors. The red dashed vertical line indicates the lower frequency cut-off used in calculating Faithfulness values.

The faithfulness calculation is done numerically, and therefore, the integral (3.10) is replaced with a sum. The waveforms and noise PSD are discrete data sets of equal length, which is managed by cutting them off at 22

the same minimum frequency of 20 Hz and the same maximum frequency, determined by whichever one of the waveforms has the lowest. The waveforms are generated with code built into LALSuite, and only one of the polarisation, h+ , is used. The h× polarisation is the same as h+ with possibly a different overall amplitude (depending on the inclination of the binary), which would be lost in normalising the waveform. It therefore does not contain any additional information for the overlap. Since the SEOBNRv3 code produces a time domain waveform, its output is transformed with a fast Fourier transform (FFT). Before the FFT, a tapered window is added to the waveform at frequencies below 20 Hz, to avoid sharp cut-offs. The SEOBNRv2ROM and IMRPhenomPv2 waveforms are directly generated in the frequency domain. For the maximisation of the product, both waveforms are generated once with the given parameters. Afterwards, the external parameters φ0 and t0 are changed by multiplying the waveform with a factor e2iφ0 and an array [e−2iπf t0 ] for each frequency f , respectively. For the optimisation, the python routine optimize.fmin from the Scipy module is used (a minus sign is introduced in the faithfulness function so that it can be minimised instead of maximised).

3.2.2

Probing parameter space

For the parameterised test of GR on GW150914 and GW151226, the waveform IMRPhenomPv2 has been used. In order to back up the GR tests, IMRPhenomPv2 would only need to be checked around the binary parameters of both events. The total masses and mass ratios are quite well-measured (M = 64.8+4.6 −3.9 M , +0.42 +5.9 +2.43 q = 1.22−0.20 and M = 21.8−1.7 M , q = 1.92−0.83 respectively, (Abbott et al., 2016c)), but this is not true for most other parameters. Besides this limitation, a test that covers a larger part of parameter space is also more desirable than a more restricted test. It would provide extended information on the waveforms themselves, an also possibly be of use when new gravitational waves are measured. In the most general case, the intrinsic parameter space of coalescing binaries is eight dimensional: two masses and two spins vectors with three components each. The position and the orientation of the system relative to the observer add another four parameters. It is impossible to vary all of these parameters at the same time, so a feasible test will have to be restricted in some way. It is therefore useful to investigate which parameters have any, or the most significant, effect on the waveform. For the extrinsic parameters, only the inclination of the binary system can affect the shape of the waveform. The sky position of the source determines the angle at which the wave passes through the detector but this only affects the relative strength of the two waveform polarisations, not their shape. Only the latter is considered in this work, hence a single polarisation is used and the sky position is irrelevant. This is also true for the distance, which is responsible for an overall factor in the amplitude. This information is lost in the faithfulness calculation, since the waveforms are normalised. The inclination only has an effect on waves from binaries with precessing spins; the amplitude and phase modulations caused by precession are weakest for face-on and strongest for edge-on systems Schmidt et al. (2015). On the other hand, most of the intrinsic parameters of the binary are important for the shape of the waveform. For non-precessing spins, there are four: m1 , m2 , χ1 and χ2 , where both spins are of course aligned with the orbital angular momentum. An equivalent description of the two component masses is a combination of the total mass M and the mass ratio q, which is used in this work. Precessing spins need three parameters each, which are chosen to be the magnitude χi , the angle with the orbital angular momentum θi and the angle of the in-plane projection of the spin with the x-axis φi (for an overview of the angles involved, see Fig. 3.2). The first two are clearly relevant parameters; their combination determines the size of the spin ~ and perpendicular to L. ~ In the phenomenological models, these are reduced to components aligned with L two parameters χe from the aligned and χp from the perpendicular components, and appear to be the final measurable parameters (Khan et al., 2016; Schmidt et al., 2015). However, these are approximations and in general all four spin components are relevant. On the other hand, both in-plane angles φi may not be relevant at all. The black holes orbit each other, and there is no fixed start time (it is maximised over in the faithfulness calculation), so the direction of the x-axis can be chosen arbitrarily. Then, both angles φ1 and φ2 reduce to their difference, φ = |φ1 − φ2 |. Furthermore, the in-plane spin-components rotate due to precession, such that φ changes constantly. Recall

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~ is the orbital Figure 3.2: Overview of the angles involved with the spin and angular momentum vectors. L ~ ~ angular momentum of the binary, S1 the spin of the bigger black hole and S2 that of the smaller. Vector sizes are chosen arbitrarily in this figure. that this is what the χp approximation is built upon (see section 3.1.2). If the spins rotate at the same rate however - which is the case for equal mass equal spins systems - the initial value for φ is maintained. Only in those configurations, φ is relevant as is determines the strength of the main precession effect, namely spin-orbit coupling at 1.5PN order (see Eq. (3.6)). Identifying all the possibly relevant parameters yields the following list: M , q, χ1 , χ2 , θ1 , θ2 , φ and ι (see also Table 3.2.2). Even for non-precessing spins, which require only the first four parameters, this is too much to vary over a large range at the same time, let alone visualise in a two-dimensional plot. Focusing on the parameters of GW150914 or GW151226 allows for a choice of the masses and mass ratios. Respecting the uncertainty in these estimated parameters, the values at the upper and lower 90% confidence intervals (CI) are used as well (Abbott et al., 2016a). For GW150914, an extra mass ratio value of 1.43 is added to fill the gap between the estimated value of 1.22 and the upper 90% CI of 1.64. A similar thing is done for GW151226, but for both the upper and the lower 90% CI since the uncertainties are rather large. Having a few fixed values for the masses, the spin magnitudes remain as the only relevant variables for the non-precessing case. The dimensionless spins χi = |S~i |/m2i have a physically limited range from zero to one, so they are easy to use as variables. To use both directions of the spin (one with a component aligned ~ and one with a component anti-aligned with L), ~ the range is extended from minus one to one. χ1 is with L chosen as the x-variable and χ2 as the y-variable, and both ranges of [-1, 1] are probed at 50 points, resulting in a total of 50 × 50 = 2500 faithfulness calculations per test. These points can be visualised as a colour plot with χ1 and χ2 on the x- and y-axes, respectively. The same is done for the precessing spin configuration, where a choice needs to be made for the remaining parameters θ1 , θ2 , φ and ι. The faithfulness is a somewhat computationally expensive calculation, since both waveforms need to be calculated for each set of parameters, and the value needs to be optimised over t0 and φ0 . Besides, the number of resulting plots will multiply for the number of choices of each remaining parameter, so too many choices will clutter the results. Therefore, the choices for θ1 , θ2 are restricted to be equal, and vary over the values π/6, π/3, π/2, such that the precession effects are of different strength for each choice. Likewise, two values for the inclination are chosen, 0 and π/4, where the precession effects should be more visible for the second than for the first choice. As the φ parameter will only be relevant (at least to leading order) for equal mass, equal spin systems, it is less interesting to vary this parameter and so only the value φ = π is used, meaning that the two in-plane components of the spins point in opposite direction at the start time t0 . All discussed choices for parameters are summarised in Table 3.2.2.

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parameter M - total mass (M ) q - mass ratio (m1 /m2 > 0)

non-precessing spins yes yes

precessing spins yes yes

values for GW150914 test 60.9, 64.8, 69.4 1.0, 1.22, 1.43, 1.64

χ1 - magnitude spin 1 χ2 - magnitude spin 2 ~ of spin 1 θ1 - angle with L ~ of spin 2 θ2 - angle with L φ - in-plane spin angle ι - inclination

yes yes no no no no

yes yes yes yes in some cases yes

x-variable: [-1, 1] y-variable: [-1, 1] 0, π/6, π/3, π/2 0, π/6, π/3, π/2 π 0, = π/4

values for GW151226 test 20.1, 21.8, 27.7 1.09, 1.51, 1.92, 3.14, 4.35 x-variable: [-1, 1] y-variable: [-1, 1] 0, π/6, π/3, π/2 0, π/6, π/3, π/2 π 0, π/4

Table 3.3: Parameters used in the overlaps test of IMRPhenomPv2. The second and third columns indicate whether the parameter is relevant for respectively a configuration with non-precessing or precessing spins. The fourth and fifth columns list the values for each parameter that have been used in the test. For the total mass and mass ratio, the estimated median values for GW150914 and GW151226 are in bold font, while the other values are based on the 90% confidence intervals (Abbott et al., 2016c,a). Since no final results for the precessing spin configurations for GW151226 were obtained, these values are shown in gray.

3.2.3

Results

Non-precessing spins Both the SEOBNRv2ROM and the SEOBNRv3 model can produce waveforms from non-precessing spin configurations, so both are used in a comparison with IMRPhenomPv2 (the ROM version of SEOBNRv2 is used since it is faster). First, the results of the studies with GW150914 parameters are shown in Fig. 3.3 and 3.4, followed by the ones for GW151226 in Fig. 3.5 and 3.6. In each study, all combinations of choices for M and q (given in Table 3.2.2) are used, resulting in twelve plots for GW150914 and fifteen for GW151226. The plots are ordered by increasing M from left to right and increasing q from bottom to top. Precessing spins Overlap studies with precessing spin configuration were performed using the waveforms IMRPhenomPv2and SEOBNRv3. No final results were obtained for the studies with mass parameters of GW151226, so only results for GW150914 are presented here. Studies with an orbital inclination of zero are shown first, with ~ of π/6 in Fig. 3.7, π/3 in Fig. 3.8 and π/2 in Fig. 3.9. Results angles between the spin components and L for configurations with the same precession angles but with an inclination of π/4 are shown in Fig. 3.10, 3.11 and 3.12. The individual plots with different values for M and q are ordered in the same way as for the non-precessing spin studies.

3.2.4

Discussion

Meaning of the Faithfulness scale In the results in general, faithfulness values vary drastically from close to 1.0 to low values of around 0.7. In order to interpret these results, it is useful to provide some context. Concerning the accuracy of the parameterised test (or other data analysis purposes), what finally matters is by how much the inaccuracy of the waveform affects the result, compared to for example the error due to noise. Abbott et al. (2016d) report, besides the parameterised test of GR on GW150914, properties of the residuals of the data after subtracting the highest likelihood model waveform. An upper bound on the signal to noise ratio (SNR) in the residuals of SNRres ≤ 7.3 (at 95% confidence) was found, consistent with SNR found in noise at times close to the signal. Hypothetically, if there is no noise, a residual SNR can be calculated from having a mismatch between the

25

model and the signal waveforms alone, given by SNR2res = (1 − E 2 ) E −2 SNR2det (Abbott et al., 2016d), where E is the effectualness between the model and signal (see Section 3.2.1). Filling in the measured SNRres ≤ 7.3 and SNRdet = 23.7, an lower bound on the match between the highest likelihood model waveform and that of the underlying signal can be found, which is E ≥ 0.96. Therefore, one could say that a mismatch of 1 − E ≤ 0.04 in a model waveform is not measurable over the noise, which makes that model suitable for the parameterised test. However, this uses the effectualness as a measure, for which the faithfulness is a lower limit (see section 3.2.1), and so somewhat lower matches are probably acceptable as well. In this study, model waveforms are not compared with NR waveforms, but with each other. If the overlaps between two model waveforms is sufficiently high, presumably, their respective overlaps with NR would also be, since both models are based on NR calibrations. In short, F ≥ 0.96 seems to be a reasonable criterion for the waveform models, although waveforms with somewhat lower overlaps are likely suitable as well. Additionally, a scale for faithfulness values can be set by the matches of waveform models with NR waveforms, preferably ones that were not used in its calibration. Taracchini et al. (2014) report matches of F ≥ 0.99 for SEOBNRv2 with NR waveforms in its calibration region, and F ≥ 0.96 with two precessing NR waveforms. (The first with q = 3.0, χ1 = 0.5 at an angle of 0.499π and χ2 = 0.499 at an angle of 0.987π with the orbital angular momentum. The second with q = 5.0, χ1 = 0.499 at an angle of 0.499π and χ2 = 0.) The same two waveforms were used to assess SEOBNRv3, which has a match F & 0.99 for the first and F & 0.98 for the second (see Fig. 10 in Pan et al. (2014)). Using the ROM version of SEOBNRv2 can introduce an additional error, but SEOBNRv2ROM has overlaps F ≥ 0.99 against SEOBNRv2 Khan et al. (2016). It seems that matches of around 0.99 (or somewhat lower) should be expected within the calibration region of SEOBNRv2(ROM)/SEOBNRv3. The IMRPhenomD model has been compared to all of its calibration NR waveforms, and an additional 28 waveforms inside its calibration region. Most matches were found to be around F ∼ 0.999, and almost all were above F > 0.99, while the worst was F > 0.96 Khan et al. (2016). Additionally, Khan et al. (2016) studied overlaps between IMRPhenomD and SEOBNRv2, in which they found quite a wide range of overlaps, but these will be discussed along with the results of this work in the following section 3.2.4. For the IMRPhenomPv2 model, no comparisons with NR waveforms were made, but the validity of the effective precession spin approximation was verified with overlaps between inspiral waveforms with and without the approximation. It was found that only in edge cases with two large in-plane spin components, overlaps were below F . 0.96, predominantly due to higher order spin-spin effects not included in the final model Schmidt et al. (2015).

Non-precessing spins For the non-precessing spin cases, there are overlaps results between IMRPhenomPv2 and SEOBNRv2ROM as well as between IMRPhenomPv2 and SEOBNRv3, and they are rather comparable to each other. This is to be expected, since the SEOBNRv3 model uses SEOBNRv2 as a basis to add precession effects to. So for non-precessing configurations, they should output the same waveforms. Minor differences between the results can be attributed to different numerical implementations of the same model. This is also true of the IMRPhenomPv2 model, which for non-precessing spins reduces to the IMRPhenomD model. For what follows, the results can be considered simply as the match between the IMRPhenomD and the SEOBNRv2(ROM) models, and the results of the next section should inform on the influence of the precession models. For the study with parameters close to GW150914 (Fig. 3.3 or 3.4), large parts of the parameter space slices have very high faithfulness values F > 0.997, and almost the entire slices have F > 0.96. For the equal mass cases, no matches below 0.96 are found, and only get close to 0.96 at the highest positive spin values. The somewhat lower matches in these regions can be explained on the basis of the set of NR waveforms used to calibrate IMRPhenomD (see Table 3.1.2) and the one used to calibrate SEOBNRv2 (see Table 3.1.1); there are no waveforms included where on of the spins is high and positive, and one is high and negative. The same effect can be observed in the non-equal mass slices of parameter space, where matches do fall slightly below F = 0.96 in the bottom right corners. However, here, even lower faithfulness values of 26

F . 0.9, are found in the upper right corners, where both spins are high and positive. Again, this is very likely due to the lack of calibration NR waveforms at these parameter values. For IMRPhenomD, waveforms with q = 4.0 and spins up to 0.75, and with q = 8.0 and spins up to 0.85 were used, but none higher than 0.9 except for the equal mass case. For SEOBNRv2, no spin values higher than 0.5 were used with q > 1.0. Since neither model is fully calibrated at these points, it is impossible to say which one, if not both, are incorrect. The high faithfulness values at high positive spins found in the equal mass slices suggest that additional calibration waveforms may very well be able to fix the discrepancy, since both calibration sets do include high spin configurations with q = 1 (A5 in Table 3.1.2 for IMRPhenomD and 9 in Table 3.1.1 for SEOBNRv2). These results are in excellent agreement with the overlaps between IMRPhenomD and SEOBNRv2 for equal spin configurations studied by Khan et al. (2016), who found (for masses M ≥ 50 M ) high faithfulness matches except for χ1 = χ2 & 0.6 and q > 1.0. On the other hand, for high negative spins, the the faithfulness values only decrease slightly compared to those at lower spins. This might be due to orbital hang-up (Campanelli et al., 2006); the effect that for ~ slow the orbit down, while anti-aligned spins speed it up, rapidly spinning black holes, spins aligned with L compared to the non-spinning case. This means that the inspiral part of the waveform for negative spins is shorter (relative to the merger-ringdown) than for positive spins, and therefore has a lower weight in the total overlap. It has been verified that this effect occurs in all the waveform models that were used. Khan et al. (2016) already found that for waveforms with relatively longer inspirals - namely, with lower masses the match between IMRPhenomD and SEOBNRv2 was worse than for waveforms with shorter inspirals. This is due to the inspiral waveform used to build IMRPhenomD, which is an uncalibrated version of SEOBNRv2 and hence somewhat different. Indeed, the effect of a longer inspiral is also found in the difference between the GW150914 and the GW151226 overlaps; the latter has a lower total mass and so a longer inspiral, which explains the generally lower matches in its plots compared to GW150914. Taking a closer look at Fig. 3.5 (or 3.6), the results for GW151226 show some of the same effects as those for GW150914. Again, the lowest matches are found at high positive spins, with q > 1.0, and the effect is worse for the highest mass ratios. For the mass ratios up to 1.92, the corners with one high positive and one high negative spin have matches F < 0.96, similar to the GW150914 results. The effect is stronger, which is likely due to the increase in inspiral length. For the slices with mass ratios q = 3.14 and q = 4.35 on the other hand, low matches are found where |χ1 | & 0.6, and the size of χ2 seems to matter less. Naturally, for greater mass ratios, the relative influence of the more massive black hole becomes stronger, and is more dominant in determining the size of χP N (see Eq. (3.5)), which drives the leading order spin effect during inspiral. Finally, the plots for GW151226 seem much smoother than those for GW150914, which is probably another effect of the longer inspiral waveforms, which makes the faithfulness values more accurate.

Precessing spins For precessing spin configurations, final results were obtained only for GW150914 parameters. For the study with GW151226, many points had very low matches, not in line with the rest of the points on the slice. It may be that the optimisation in the faithfulness calculation did not yield the correct results, or erroneous waveforms were produced. Reliable results could not be obtained and are therefore omitted in this work. Additionally, it was found that the code for SEOBNRv3 generates faulty waveforms for configurations with equal mass and equal spins. This was resolved by adjusting one of the spins with 1 × 10−5 at these points. Some points with lower overlaps can still be found on the line of equal spins in the q = 1.0 plots. In general, high overlap values can be found on each parameter slice. Specifically for |χ1 | ≤ 0.5 and |χ2 | ≤ 0.5 at least, overlaps are F ≥ 0.96 in all cases, indicating that the waveform models are suitable for the parameterised test. For many slices, this validated region extends to higher spins. Regions with F < 0.96 at high spins get larger for an increasing angle θ of the spins with the orbital angular momentum, or an increasing inclination. In both cases, the overlaps are lower when precession effects are stronger. As with the non-precessing configurations, lower overlaps are found for larger spins. At least part of this result is due to the lower matches in the underlying non-precessing models themselves 27

(due to lack of calibration NR waveforms). Additionally, precession effects are stronger for larger spins, and it was already observed that stronger precession seems to result in lower matches. This indicates that the approximations made in either IMRPhenomPv2 or SEOBNRv3 or both, are not entirely accurate. The effective precession parameter χp used in IMRPhenomPv2 only captures the leading order precession effect, ~1 · S ~2 becomes larger, and hence the namely spin-orbit coupling, and ignores spin-spin coupling. Naturally, S approximation worse, for larger spins. For both models, no additional precessing NR waveforms were used for calibration, and so it is not surprising that the models differ more when adding stronger effects on top of the underlying calibrated models. In contrast to the overlaps with non-precessing waveforms, the lowest matches are no longer found in the region with two high positive spins. This may be due to the spins being at an angle θ with the orbital angular momentum, such that the aligned spin components are reduces by a factor cos θ. Then, the parameters of the underlying non-precessing model are closer to, or even within its calibration region. For the studies with θ = π/6 or θ = π/3, very low overlaps of F < 0.9 are found in regions where χ1 is high and positive and χ2 is high and negative (the size of the regions differ, but nowhere is F < 0.9 for |χi | < 0.5). For the studies with completely in-plane spins, F < 0.9 can also be found in the bottom right corners, but also in the opposite corner where χ1 is high and negative and χ2 is high and positive. Both regions are smaller for the highest two mass ratio’s, q = 3.14 and q = 4.35.

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Faithfulness of IMRPhenomPv2 and SEOBNRv2ROM GW150914, non-precessing spins, inclination = 0

Figure 3.3: Faithfulness of the waveforms IMRPhenomPv2and SEOBNRv2, with non-precessing spins and zero inclination. Plots have different values for the total mass and mass ratio, chosen around the measured values of GW150914 (see Table 3.2.2), with M increasing from left to right and increasing q from bottom to top. Each plot has the spins of object 1 (the most massive) on the x-axis and the spin of object 2 on the y-axis. Negative spins correspond to a spin vector anti-aligned with the orbital angular momentum. The faithfulness values are shown with a colourmap ranging from 0.7 to 1.0, with contours indicating the values 0.96, 0.99 and 0.996. A dashed gray line indicates the points where the effective spin parameter χe = 0.

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Faithfulness of IMRPhenomPv2 and SEOBNRv3 GW150914, non-precessing spins, inclination = 0

Figure 3.4: Faithfulness of the waveforms IMRPhenomPv2and SEOBNRv3, with non-precessing spins and zero inclination. Plots have different values for the total mass and mass ratio, chosen around the measured values of GW150914 (see Table 3.2.2), with M increasing from left to right and increasing q from bottom to top. Each plot has the spins of object 1 (the most massive) on the x-axis and the spin of object 2 on the y-axis. Negative spins correspond to a spin vector anti-aligned with the orbital angular momentum. The faithfulness values are shown with a colourmap ranging from 0.7 to 1.0, with contours indicating the values 0.96, 0.99 and 0.996. A dashed gray line indicates the points where the effective spin parameter χe = 0.

30

Faithfulness of IMRPhenomPv2 and SEOBNRv2ROM GW151226, non-precessing spins, inclination = 0

Figure 3.5: Faithfulness of the waveforms IMRPhenomPv2and SEOBNRv2, with non-precessing spins and zero inclination. Plots have different values for the total mass and mass ratio, chosen around the measured values of GW151226 (see Table 3.2.2), with M increasing from left to right and increasing q from bottom to top. Each plot has the spins of object 1 (the most massive) on the x-axis and the spin of object 2 on the y-axis. Negative spins correspond to a spin vector anti-aligned with the orbital angular momentum. The faithfulness values are shown with a colourmap ranging from 0.7 to 1.0, with contours indicating the values 31points where the effective spin parameter χe = 0. 0.96, 0.99 and 0.996. A dashed gray line indicates the

Faithfulness of IMRPhenomPv2 and SEOBNRv3 GW151226, non-precessing spins, inclination = 0

Figure 3.6: Faithfulness of the waveforms IMRPhenomPv2and SEOBNRv3, with non-precessing spins and zero inclination. Plots have different values for the total mass and mass ratio, chosen around the measured values of GW151226 (see Table 3.2.2), with M increasing from left to right and increasing q from bottom to top. Each plot has the spins of object 1 (the most massive) on the x-axis and the spin of object 2 on the y-axis. Negative spins correspond to a spin vector anti-aligned with the orbital angular momentum. The faithfulness values are shown with a colourmap ranging from 0.7 to 1.0, with contours indicating the values 0.96, 0.99 and 0.996. A dashed gray line indicates the 32points where the effective spin parameter χe = 0.

Faithfulness of IMRPhenomPv2 and SEOBNRv3 GW150914, spins precessing at π/6, inclination = 0

Figure 3.7: Faithfulness of the waveforms IMRPhenomPv2and SEOBNRv3, with spins precessing at an angle of π/6 with the orbital angular momentum, and zero inclination. Plots have different values for the total mass and mass ratio, chosen around the measured values of GW150914 (see Table 3.2.2), with M increasing from left to right and increasing q from bottom to top. Each plot has the spin magnitude of object 1 (the most massive) on the x-axis and the spin of object 2 on the y-axis. Negative spins correspond to a spin ~ and a negative vector in the opposite direction, such that a positive spin has a component aligned with L ~ spin has a component anti-aligned with L. The faithfulness values are shown with a colourmap ranging from 0.7 to 1.0, with contours indicating the values 0.9, 0.96 and 0.99. A dashed gray line indicates the points where the effective spin parameter χe = 0.

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Faithfulness of IMRPhenomPv2 and SEOBNRv3 GW150914, spins precessing at π/3, inclination = 0

Figure 3.8: Faithfulness of the waveforms IMRPhenomPv2and SEOBNRv3, with spins precessing at an angle of π/3 with the orbital angular momentum, and zero inclination. Plots have different values for the total mass and mass ratio, chosen around the measured values of GW150914 (see Table 3.2.2), with M increasing from left to right and increasing q from bottom to top. Each plot has the spin magnitude of object 1 (the most massive) on the x-axis and the spin of object 2 on the y-axis. Negative spins correspond to a spin ~ and a negative vector in the opposite direction, such that a positive spin has a component aligned with L ~ spin has a component anti-aligned with L. The faithfulness values are shown with a colourmap ranging from 0.7 to 1.0, with contours indicating the values 0.9, 0.96 and 0.99. A dashed gray line indicates the points where the effective spin parameter χe = 0.

34

Faithfulness of IMRPhenomPv2 and SEOBNRv3 GW150914, spins precessing at π/2, inclination = 0

Figure 3.9: Faithfulness of the waveforms IMRPhenomPv2and SEOBNRv3, with spins precessing at an angle of π/2 with the orbital angular momentum, and zero inclination. Plots have different values for the total mass and mass ratio, chosen around the measured values of GW150914 (see Table 3.2.2), with M increasing from left to right and increasing q from bottom to top. Each plot has the spin magnitude of object 1 (the most massive) on the x-axis and the spin of object 2 on the y-axis. Negative spins correspond to a spin ~ and a negative vector in the opposite direction, such that a positive spin has a component aligned with L ~ The faithfulness values are shown with a colourmap ranging from spin has a component anti-aligned with L. 0.7 to 1.0, with contours indicating the values 0.9, 0.96 and 0.99. A dashed gray line indicates the points where the effective spin parameter χe = 0.

35

Faithfulness of IMRPhenomPv2 and SEOBNRv3 GW150914, spins precessing at π/6, inclination = π/4

Figure 3.10: Faithfulness of the waveforms IMRPhenomPv2and SEOBNRv3, with spins precessing at an angle of π/6 with the orbital angular momentum, and zero inclination. Plots have different values for the total mass and mass ratio, chosen around the measured values of GW150914 (see Table 3.2.2), with M increasing from left to right and increasing q from bottom to top. Each plot has the spin magnitude of object 1 (the most massive) on the x-axis and the spin of object 2 on the y-axis. Negative spins correspond to a spin ~ and a negative vector in the opposite direction, such that a positive spin has a component aligned with L ~ spin has a component anti-aligned with L. The faithfulness values are shown with a colourmap ranging from 0.7 to 1.0, with contours indicating the values 0.9, 0.96 and 0.99. A dashed gray line indicates the points where the effective spin parameter χe = 0.

36

Faithfulness of IMRPhenomPv2 and SEOBNRv3 GW150914, spins precessing at π/3, inclination = π/4

Figure 3.11: Faithfulness of the waveforms IMRPhenomPv2and SEOBNRv3, with spins precessing at an angle of π/3 with the orbital angular momentum, and zero inclination. Plots have different values for the total mass and mass ratio, chosen around the measured values of GW150914 (see Table 3.2.2), with M increasing from left to right and increasing q from bottom to top. Each plot has the spin magnitude of object 1 (the most massive) on the x-axis and the spin of object 2 on the y-axis. Negative spins correspond to a spin ~ and a negative vector in the opposite direction, such that a positive spin has a component aligned with L ~ spin has a component anti-aligned with L. The faithfulness values are shown with a colourmap ranging from 0.7 to 1.0, with contours indicating the values 0.9, 0.96 and 0.99. A dashed gray line indicates the points where the effective spin parameter χe = 0.

37

Faithfulness of IMRPhenomPv2 and SEOBNRv3 GW150914, spins precessing at π/2, inclination = π/4

Figure 3.12: Faithfulness of the waveforms IMRPhenomPv2and SEOBNRv3, with spins precessing at an angle of π/2 with the orbital angular momentum, and zero inclination. Plots have different values for the total mass and mass ratio, chosen around the measured values of GW150914 (see Table 3.2.2), with M increasing from left to right and increasing q from bottom to top. Each plot has the spin magnitude of object 1 (the most massive) on the x-axis and the spin of object 2 on the y-axis. Negative spins correspond to a spin ~ and a negative vector in the opposite direction, such that a positive spin has a component aligned with L ~ spin has a component anti-aligned with L. The faithfulness values are shown with a colourmap ranging from 0.7 to 1.0, with contours indicating the values 0.9, 0.96 and 0.99. A dashed gray line indicates the points where the effective spin parameter χe = 0.

38

Chapter 4

Reliability and sensitivity of the parameterised test In the previous Chapter, the method of the parameterised test of GR has been investigated by verifying the waveform model that was used (IMRPhenomPv2). In the following, two different tests are discussed that are concerned with the interpretation of the results of the parameterised test, specifically those for GW150914. The first section contains a description and preliminary results of an investigation on the effect of detector noise on the measured posteriors. In the second, the sensitivity of the parameterised test is probed by running it on data containing a model signal with GR deviations, for which partial results are presented as well.

4.1

The effect of noise

The results of the parameterised test on GW150914 show offsets from zero in the deviation coefficients of the post-Newtonian parameters of around 2–2.5σ (Abbott et al., 2016a), as can be seen in the top panel of Fig. 2.1. However, posteriors for GW151226 are all well-centered on zero, as are those for the intermediate and merger-ringdown regimes of GW150914. It is expected that this difference stems from the short length of the inspiral part of GW150914 that was in band for the LIGO detectors. Only ∼ 8 inspiral cycles are measured above 30 Hz (some of which fall in the intermediate regime), whereas for GW151226, there are roughly ten times as many (see Fig. 1.2). Therefore, the measurements on the inspiral of GW150914 were more susceptible to noise than the other measurements, which could explain the offsets. In order to test this, a numerical relativity (NR) waveform with masses and spins as estimated for GW150914 is produced, and injected into fifteen different segments of data containing only noise. The masses that are used are m1 = 40.83M , m2 = 33.26M , and the spins vectors are χ ~ 1 = (0.092, 0.038, 0.326) and χ ~ 2 = (0.215, 0.301, −0.558). Then, the parameterised test is run on these data segments. The stretches of data are taken from LIGO’s S6 run, which ran from July till 2009 October 2010, and recoloured to follow the predicted noise curve for Advanced LIGO, as plotted in Fig. 3.2.1. Each noise segment was checked to not contain any glitches, since these could also disturb the measurement, which is not the case for GW150914.

4.1.1

Results

The GW150914-like NR waveform has been injected in fifteen different stretches of data. On each, the parameterised test was run for each of the deviation parameters of the inspiral region of the IMRPhenomPv2 model. These are {δ χ ˆ0 , δ χ ˆ1 , δ χ ˆ2 , δ χ ˆ3 , δ χ ˆ4 , δ χ ˆ5l , δ χ ˆ6 , δ χ ˆ6l , δ χ ˆ7 } (see the inspiral parameters {φi } in appendix A in Khan et al. (2016)). Not all the runs are finished at the time of writing, but results for those that are, are presented here. Table 4.1 provides some statistics on the runs, namely the median and standard

39

deviation for each posterior of the diviation parameters, the significance calculated as the ratio of these, and the recovered signal to noise ratio. Full violin plots of the posteriors can be found in Appendix A.

4.1.2

Discussion

Of all the results, one posterior probability distribution is found to have an offset of more than 2σ, namely that of δ χ ˆ1 for noise 1, with a median that’s 2.07σ away from zero. More posteriors are found to be offcentered, but those are sufficiently wide for the offset to be in the range 1–2σ. Thus, it seems that the 2–2.5σ offsets found in the posteriors of the inspiral parameters of GW150914 are outliers. However, most of the signals are recovered with significantly higher SNR than that of GW150914, which was 23.7 (Abbott et al., 2016a). Only noise 15 has an SNR of around this level, and noise 7 a somewhat lower SNR. Noise 9 has very low SNR and very wide, uninformative posteriors. It is expected that measurements with higher signal to noise ratio are less affected by the noise fluctuations, and thus have smaller offsets. Two noise realisations are not enough to inform on the distribution of offsets, or specifically the rarity of 2.5σ offsets. To do this test properly, the SNR would have to be controlled to be around the level of that of GW150914. What affects the SNR are the total mass of the binary, its distance to the detector and the angle at which the signal passes through the detector. The latter is determined by the sky position of the source relative to the position of the detector on earth at the time of the event. In order to obtain conclusive results for this test, then, these parameters need to be carefully controlled, for example by recreating those of GW150914. So far, the results of the test do suggest that offsets in deviation coefficients such as measured for GW150914 can occur due to different noise realisations, only it will have to be confirmed by further tests that this happens at the 2–2.5σ level for SNR ∼ 23.7. Apart form the size of the offsets, it is also notable that the posteriors for the inspiral deviation coefficients from GW150914 are (almost) alternatingly shifted to positive and negative values. While the deviation coefficients in the set {δ χ ˆ1 , δ χ ˆ3 , δ χ ˆ5l , δ χ ˆ6l , δ χ ˆ7 } have positive offsets, the others in {δ χ ˆ0 , δ χ ˆ2 , δ χ ˆ4 , δ χ ˆ6 } have negative ones. The same behaviour is observed in the posteriors from this investigation. See Section 4.2.2 for a discussion on measuring a disturbance in a waveform, in this case due to noise, with different deviation parameters. Besides redoing this test with controlled SNR for GW150914, it would be interesting to see the same test on a waveform with parameters like GW151226. This is being worked on; unfortunately, these runs take longer as the waveforms have many more cycles (the reason for doing this test in the first place). It is very much expected that these results would have well-centered posteriors as in the original parameterised test on GW151226, which would be strong evidence for the difference being due to inspiral length. For the same reason, the test could be extended to include the merger-ringdown coefficients of GW150914, since the entire merger-ringdown regime is in band for the detector and the actual parameterised test has well-centered posteriors here as well.

4.2

Sensitivity of the parameterised test

With the results of the parameterised test of GR, deviations that are outside the support of the measured posteriors can be excluded. It is an interesting question to ask whether a signal containing a deviation beyond these limits would have resulted in a measurement that refutes GR. In general, it would be good to know how strong a deviation must be for it to be detected to ascertain the sensitivity of the parameterised test. This is done by running the test on data containing a signal with a built-in deviation from GR. A non-GR signal can not be produced with numerical relativity, since this simulates spacetime completely according to GR. Instead, a waveform is made with the IMRPhenomPv2 model, using the deviation coefficients from the parameterised test to introduce non-GR shifts. This is done by setting one of the deviation coefficients to a non-zero value, such that the waveform is changed by φi → (1 + δ φˆi ) φi . Particularly, the size of δ φˆi is chosen at the edge of the posteriors measured with GW150914, at five times the standard deviation. This should be the value at which a deviation would have been detected (with a 5σ significance), but it is

40

median -0.04 0.43 -0.13 0.11 -0.73 0.35 -0.58 3.79 1.70

noise 1 stdev significance 0.04 1.00 0.21 2.07 0.13 1.01 0.10 1.14 0.73 1.00 0.26 1.32 0.48 1.22 2.72 1.39 1.26 1.35

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7

median 0.04 -0.11 0.14 -0.08 0.88 -0.30 0.57 -3.41 -1.41

noise 2 stdev significance 0.05 0.95 0.21 0.50 0.12 1.09 0.09 0.90 0.57 1.53 0.27 1.11 0.49 1.19 2.77 1.23 1.26 1.12

SNR 40.44 40.46 40.46 40.45 40.57 40.49 40.49 40.54 40.52

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7

median -0.04 0.10 -0.11 0.07 -0.58 0.24 2.34 1.00

noise 3 stdev significance 0.03 1.16 0.18 0.56 0.10 1.18 0.06 1.02 0.52 1.10 0.21 1.12 2.15 1.09 0.93 1.08

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7

median -0.04 0.06 -0.15 0.09 -0.67 -0.54 3.46 1.51

noise 4 stdev significance 0.03 1.21 0.16 0.38 0.10 1.57 0.07 1.32 0.50 1.34 0.40 1.34 2.32 1.49 1.02 1.48

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7

median -0.01 0.04 0.01 -0.00 0.10 -0.03 0.10 -0.10 -0.33

noise 5 stdev significance 0.03 0.26 0.17 0.22 0.09 0.10 0.07 0.04 0.48 0.21 0.17 0.17 0.33 0.29 2.14 0.05 0.84 0.39

SNR 59.27 59.30 59.29 59.29 59.32 59.30 59.32 59.34 59.33

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7

median -0.02 0.27 -0.07 0.04 -0.37 0.17 -0.27 1.63 0.71

noise 6 stdev significance 0.03 0.84 0.14 1.92 0.09 0.82 0.06 0.67 0.47 0.78 0.18 0.90 0.32 0.84 1.95 0.83 0.90 0.78

SNR 60.48 60.52 60.50 60.49 60.53 60.51 60.52 60.56 60.54

SNR 55.59 55.61 55.61 55.60 55.64 55.62 55.67 55.65

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7

median -0.07 0.37 -0.25 0.15 -1.40 0.57 -1.00 5.67 2.91

noise 7 stdev significance 0.11 0.64 0.39 0.94 0.25 0.99 0.15 1.02 1.26 1.11 0.46 1.24 0.86 1.17 5.08 1.12 2.30 1.26

SNR 20.13 20.23 20.19 20.18 20.28 20.24 20.27 20.36 20.32

SNR 55.30 55.32 55.32 55.31 55.36 55.35 55.38 55.37

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7

median 0.08 0.14 0.26 -0.17 1.32 -0.51 0.91 -5.58 -2.38

noise 8 stdev significance 0.05 1.54 0.29 0.49 0.16 1.67 0.10 1.64 0.83 1.58 0.31 1.67 0.56 1.62 3.46 1.61 1.53 1.56

SNR 33.35 33.36 33.38 33.37 33.42 33.40 33.42 33.47 33.44

SNR 38.63 38.71 38.67 38.66 38.71 38.69 38.71 38.75 38.74

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7

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median -0.09 -0.48 0.17 1.27 -0.45 1.08 -5.99 -3.60

noise 9 stdev significance 7.04 0.01 3.08 0.15 5.42 0.03 5.76 0.22 5.36 0.08 5.73 0.19 7.82 0.77 6.47 0.56

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7

median 0.01 0.40 0.03 -0.02 -0.02 -0.02 -0.03 0.17 0.16

noise 11 stdev significance 0.04 0.34 0.22 1.82 0.14 0.25 0.08 0.25 0.65 0.03 0.26 0.09 0.49 0.06 3.17 0.05 1.37 0.12

SNR 39.46 39.53 39.49 39.47 39.52 39.50 39.51 39.56 39.54

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7

median 0.01 -0.03 0.03 -0.01 0.03 -0.01 0.01 -0.20 0.10

noise 12 stdev significance 0.04 0.33 0.20 0.13 0.12 0.24 0.07 0.16 0.58 0.05 0.24 0.05 0.37 0.04 2.58 0.08 1.09 0.09

SNR 50.38 50.41 50.40 50.39 50.43 50.42 50.42 50.46 50.45

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7

SNR 7.32 7.46 7.42 7.60 7.49 7.56 7.75 7.69

median 0.02 -0.01 0.07 -0.04 0.34 -0.10 0.21 -1.34 -0.57

noise 13 stdev significance 0.04 0.56 0.16 0.04 0.11 0.69 0.07 0.50 0.58 0.58 0.21 0.50 0.41 0.52 2.35 0.57 1.07 0.53

SNR 48.26 48.29 48.28 48.27 48.32 48.29 48.31 48.35 48.33

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7

median -0.02 0.35 -0.10 0.08 -0.55 0.20 -0.43 2.13 1.11

noise 14 stdev significance 0.05 0.44 0.27 1.30 0.16 0.60 0.11 0.75 0.82 0.67 0.30 0.65 0.59 0.73 3.46 0.61 1.50 0.74

SNR 32.91 32.99 32.96 32.94 33.01 32.97 32.99 33.04 33.02

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7

median -0.01 -0.12 -0.06 0.04 -0.31 0.10 -0.16 1.19 0.41

noise 15 stdev significance 0.06 0.20 0.31 0.37 0.20 0.28 0.13 0.36 0.99 0.31 0.35 0.29 0.69 0.23 4.21 0.28 1.84 0.22

SNR 23.77 23.84 23.81 23.80 23.88 23.84 23.87 23.94 23.91

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7

Table 4.1: Results of the parameterised test on different noise realisations 1 – 15, except for noise 10 for which no runs are finished yet. Each table row is a different run of the parameterised test with a given recovery parameter (param), and measured mean, standard deviation and SNR. The significance is the median divided by the standard deviation.

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parameter δχ ˆ3 δχ ˆ4 δ βˆ2 δ βˆ3 δα ˆ2 δα ˆ4

GW150914 mean 0.075 -0.702 0.115 0.084 -0.071 -0.079

GW150914 stdev 0.086 0.660 0.140 0.159 0.265 0.317

shift ± 0.4 ± 3.3 ± 0.7 ± 0.8 ± 1.3 ± 1.6

Table 4.2: Deviations from GR used in the model waveforms for the sensitivity test. The shift is the value of the given deviation parameter, used both positive and negative. Indicated are the measured mean and standard deviation for GW150914; the shift is five times the latter. in principle possible that the sensitivity of the test depends on the values of the deviation coefficients. Also, the noise can affect the measurement, as was observed with the test in the previous section 4.1. Since running the parameterised test on many different non-GR waveforms is rather expensive, a few deviation parameters are chosen to be used, mainly those that are well-measured with the original test. For each, two non-GR waveforms are generated, one with a positive and one with a negative shift. See Table 4.2 for an overview of these parameters and the size of the shifts. The other model parameters from GR are chosen as the estimated parameters for GW150914, with the same values as in the noise investigation given in Section 4.1. The test is also being done with GR parameters like GW151226 and LVT151012, but no results are available at the time of writing. Once the non-GR waveforms have been produced, they are injected in noise from the S6 run of LIGO, which is recoloured to the noise spectrum of Advanced LIGO (see also the previous test 4.1). The parameterised test is run on each of these data segments, in the same way as was done on GW150914 and GW151226.

4.2.1

Results

This test involves many runs of the parameterised test on different injected waveforms, and not all are finished at the time of writing. Preliminary results are presented in Table 4.3, with measured median and standard deviation for each posterior, along with the ratio of the latter two quantities as the significance, and the recovered SNR. Violin plots of the posteriors are given in Appendix B.

4.2.2

Discussion

The results show that, first of all, whenever a parameter is shifted in the injected waveform, this shift is measured. Furthermore, the shift is also measured with other deviation parameters (which are different runs of the parameterised test), which is expected. Take, for example, an injected waveform with a shift in δ χ ˆ3 , and call this the signal. This waveform will be changed in a certain way with respect to the GR waveform. A non-GR model where a different parameter - say δ χ ˆ2 - is allowed to vary, will unlikely match the signal best with the GR waveform where δ χ ˆ2 = 0, but probably with some other value. Indeed, it’s found that a created signal waveform with a shift of δ χ ˆ3 = 0.4 is best recovered by δ χ ˆ2 = -0.64 (if only δ χ ˆ2 is allowed to vary, that is). For the post-Newtonian coefficients, it is observed that for a certain signal, if the deviation parameters in the set {δ χ ˆ1 , δ χ ˆ3 , δ χ ˆ5l , δ χ ˆ6l and δ χ ˆ7 } are measured to be positive, the other PN parameters {δ χ ˆ0 , δ χ ˆ2 , δ χ ˆ4 and δ χ ˆ6 } are negative and vice versa. Only δ χ ˆ1 seems to not follow this rule all the time ˆ (for example at the injection with a shifted δ β3 ). This correlation was also observed in the posteriors of the noise investigation 4.1 and in the actual measurement of GW150914, as seen in section 2.2.2. A shift in one of the inspiral coefficients is not only measured with other inspiral parameters, but also to a lesser extent with intermediate and merger-ringdown parameters. Since the phase is cumulative, a shift in one of the δ χ ˆi affects the later regions too, such that δ βˆi or δ α ˆ i parameters can measure a deviation. Conversely, a shift in one of the intermediate or merger-ringdown parameters can be measured with an inspiral parameter as well, although the shift does not affect the inspiral phase. As a later part of the phase is

43

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7 δ βˆ2 δ βˆ3 δα ˆ2 δα ˆ3 δα ˆ4

median -0.21 0.65 -0.64 0.44 -3.53 1.29 -2.32 14.08 6.21 0.08 0.12 -0.10 -0.69 -0.32

δχ ˆ3 = +0.4 stdev significance 0.05 3.91 0.45 1.45 0.16 4.12 0.11 4.06 0.90 3.90 0.33 3.98 0.64 3.61 3.26 4.32 1.72 3.62 0.25 0.31 0.41 0.29 0.27 0.36 0.99 0.70 0.33 0.98

SNR 20.15 19.95 20.20 20.18 20.27 20.22 20.23 20.34 20.27 19.94 19.97 19.96 20.02 19.97

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7 δ βˆ2 δ βˆ3 δα ˆ2 δα ˆ3 δα ˆ4

median 0.20 0.08 0.62 -0.42 3.19 -1.18 2.00 -12.82 -4.96 -0.38 -0.24 0.03 0.74 0.68

δχ ˆ3 = −0.4 stdev significance 0.07 2.77 0.26 0.29 0.23 2.68 0.17 2.52 1.35 2.36 0.51 2.32 0.98 2.03 4.91 2.61 2.82 1.76 0.16 2.40 0.20 1.21 0.27 0.13 1.55 0.48 0.43 1.60

SNR 20.76 20.68 20.81 20.78 20.87 20.81 20.83 20.92 20.84 20.85 20.79 20.75 20.84 20.84

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7 δ βˆ2 δ βˆ3 δα ˆ2 δα ˆ3 δα ˆ4

median 0.22 0.11 0.75 -0.52 4.27 -1.69 3.16 -15.38 -8.87 -0.21 -0.25 -0.05 -0.59 -0.29

δχ ˆ4 = +3.3 stdev significance 0.08 2.62 0.27 0.41 0.25 2.99 0.16 3.17 1.24 3.44 0.49 3.48 0.86 3.67 3.81 4.04 2.45 3.62 0.16 1.27 0.15 1.72 0.20 0.24 1.31 0.45 0.33 0.88

SNR 22.62 22.55 22.72 22.70 22.82 22.77 22.81 22.86 22.85 22.63 22.62 22.59 22.67 22.62

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7 δ βˆ2 δ βˆ3 δα ˆ2 δα ˆ3 δα ˆ4

median -0.17 0.65 -0.54 0.37 -2.75 1.11 -2.05 12.16 5.25 0.23 -0.09 -0.22 -0.81 -0.29

δχ ˆ4 = −3.3 stdev significance 0.06 2.73 0.30 2.18 0.20 2.72 0.13 2.92 1.04 2.64 0.39 2.86 0.76 2.71 3.95 3.08 1.90 2.77 0.23 1.00 0.23 0.40 0.25 0.89 0.87 0.93 0.30 0.96

SNR 22.88 22.89 22.94 22.92 23.01 22.97 23.00 23.08 23.05 22.87 22.85 22.87 22.93 22.87

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7 δ βˆ2 δ βˆ3 δα ˆ2 δα ˆ3 δα ˆ4

δ βˆ2 = +0.8X median stdev significance -0.22 0.10 2.18 0.35 0.42 0.84 -0.71 0.31 2.29 0.41 0.19 2.17 -3.55 1.75 2.03 1.07 0.56 1.91 -2.31 1.32 1.75 12.49 5.78 2.16 5.32 2.91 1.83 0.72 0.21 3.46 0.63 0.29 2.20 -0.53 0.26 2.06 -1.40 0.56 2.50 -0.73 0.31 2.37

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7 δ βˆ2 δ βˆ3 δα ˆ2 δα ˆ3 δα ˆ4

δ βˆ2 = −0.8X median stdev significance 0.42 0.53 0.80 0.46 0.26 1.74 -0.32 0.19 1.72 2.86 1.49 1.91 -1.01 0.59 1.70 2.03 1.10 1.85 -10.76 5.02 2.14 -5.81 3.06 1.90 -0.75 0.12 6.24 -0.84 0.16 5.11 -0.27 0.24 1.16 -1.30 1.19 1.09 -0.37 0.35 1.04

SNR 21.68 21.70 21.68 21.75 21.73 21.74 21.83 21.80 22.09 21.99 21.70 21.74 21.71

SNR 22.70 22.69 22.74 22.74 22.81 22.77 22.79 22.86 22.83 23.03 22.86 22.80 22.86 22.84

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param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7 δ βˆ2 δ βˆ3 δα ˆ2 δα ˆ3 δα ˆ4

median 0.28 -0.23 -0.49 3.67 -1.29 2.49 -13.48 -6.54 -0.28 -0.28 1.05 3.94 1.21

δα ˆ 2 = +1.3 stdev significance 0.28 1.01 0.76 0.31 0.26 1.86 1.67 2.20 0.72 1.80 1.20 2.08 4.65 2.90 3.00 2.18 0.13 2.18 0.17 1.61 0.27 3.83 0.90 4.37 0.36 3.32

SNR 21.62 21.56 21.65 21.74 21.69 21.72 21.79 21.76 21.66 21.65 21.92 21.96 21.84

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7 δ βˆ2 δ βˆ3 δα ˆ2 δα ˆ3 δα ˆ4

median -1.28 -4.75 -

δα ˆ 2 = −1.3 stdev significance 0.18 7.14 0.33 14.60 -

SNR 22.16 22.13 -

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7 δ βˆ2 δ βˆ3 δα ˆ2 δα ˆ3 δα ˆ4

median 0.05 0.09 0.12 -0.08 0.49 -0.16 0.22 -1.93 -0.63 -0.26 -0.20 0.52 2.63 0.95

δα ˆ 4 = +1.6X stdev significance 0.08 0.61 0.35 0.26 0.23 0.52 0.17 0.48 1.17 0.42 0.45 0.36 0.80 0.27 4.70 0.41 2.08 0.30 0.13 2.06 0.19 1.10 0.27 1.95 1.17 2.24 0.39 2.44

SNR 20.75 20.82 20.80 20.77 20.88 20.82 20.85 20.93 20.90 20.92 20.88 20.97 21.06 21.04

param δχ ˆ0 δχ ˆ1 δχ ˆ2 δχ ˆ3 δχ ˆ4 δχ ˆ5l δχ ˆ6 δχ ˆ6l δχ ˆ7 δ βˆ2 δ βˆ3 δα ˆ2 δα ˆ3 δα ˆ4

median -4.66 -0.71 -3.54 -1.07

δα ˆ 4 = −1.6 stdev significance 4.34 1.07 0.19 3.74 0.75 4.73 0.28 3.83

SNR 22.28 22.44 22.53 22.45

Table 4.3: Results of the parameterised test on IMRPhenomPv2 model waveforms with a non-zero deviation parameter injected in detector noise. For each shifted parameter a table with results is given, insofar as they are complete. Each row is a different run of the parameterised test with a given recovery parameter (param), and measured mean, standard deviation and SNR. The significane is the median divided by the standard deviation; significances ≥ 4.0σ are in bold font.

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changed with respect to GR, a model waveform with non-zero δ χ ˆi may nevertheless better match the created signal than the exact GR waveform. Secondly, the results show a variation in significance of the recovered shifts. The shift of δ χ ˆ3 = +0.3 is recovered with a median δ χ ˆ3 = 0.44, however, the posterior is wider than with the original measurement of GW150914, and so the significance is 4.1σ. Most other inspiral parameters are also around the 4σ level, but the offsets for the intermediate and merger-ringdown parameters are much lower. The results for the shift δχ ˆ4 = +3.3 are similar, δ χ ˆ4 itself recovers the shift with 3.4σ. For the negative shifts, the significances are somewhat lower and range from 2–3σ for the inspiral coefficients, and lower for the other ones. There can be different reasons for the recovered shifts being less than the injected 5σ. Firstly, it’s notable that the recovered SNR range from ∼ 20 to ∼ 23, somewhat lower than the SNR of GW150914, which was 23.7. This may have lead to wider posteriors (comparing with Table 4.2, the runs with shifted δ χ ˆ3 and δχ ˆ4 recovered with the shifted parameter do indeed have wider posterior than those for the GW150914 measurement). Additionally, noise could have been relevant, in the same way as discussed in the previous section 4.1. Since the GR parameters of the injected model waveform are those of GW150914, it also has a short inspiral in band and is likely sensitive to noise fluctuations. Results are inconclusive about how large this effect can be, but it was already shown that 1–2σ offsets occur. This second explanation does not (or hardly) apply to the intermediate and merger-ringdown parameters, for which the entire region of the waveform is in band of the detector. The results are rather different as well. δ βˆ2 = +0.7 is best recovered with δ βˆ2 itself at 3.5σ. The negative shift however, is recovered with 6.2σ by δ βˆ2 and 5.1σ with δ βˆ3 , while the other measurements range from 0.8–1.9σ. A similar thing is observed for the shift in δ α ˆ 2 ; the positive is best measured by δ α ˆ 3 at 4.4σ, and at 3.8σ by δ α ˆ 2 itself. On the other hand, the negative shift yields hugely significant deviations with δ α ˆ 3 at 14.6σ and δ α ˆ 2 at 7.1σ. Unfortunately, here, the other runs have not finished yet. An explanation for these results could be that the sensitivity of the test varies with the value of the deviation parameter. This could be due to the structure of the parameterised deviations in the model. They enter as a fractional change of a coefficient: φi → (1 + δ φˆi ) φi . So if δ βˆ2 = +0.7, there is a pre-factor of 1.7, which is 1.7 times as large as the original factor of 1. If δ βˆ2 = −0.7, the factor becomes 0.3, which is 3.3 times as small as 1, possibly a much larger effect. This effect would be strongest around a deviation parameter of 1, which is consistent with the observations for the shift in δ βˆ2 of 0.7 and in δ α ˆ 2 of 1.3. The shifted δ α ˆ 4 = +1.6 is only recovered at 2.4 sigma by δ α ˆ 4 itself, but here too, the negative shift is measured at a higher significance of 4.7σ with δ α ˆ 3 , and at 3.8σ with δ α ˆ 4 itself. Alternatively to the implementation of the deviation parameters themselves, the sensitivity could change by other means. There is no clear reason for this to happen, however, it is in principle possible that for some shifts, the deviation from GR is better fitted with a model that’s closer to GR than for other shifts. The deviation that is measured is, in fact, only the deviation of the waveform that can not be reabsorbed by varying the GR parameters. This is described by Vallisneri (2012) as the change in the waveform orthogonal to the GR waveform. What sort of non-GR deviation is the most orthogonal to GR depends on the functional form of the deviation, and of the GR waveform model. It would require a different investigation to determine whether this is a viable explanation for the results presented here. Besides finishing this test with GW150914-like GR parameters (with shifts in other deviation parameters), it would be very interesting to see the results with model waveforms based on GW151226. Especially for the inspiral parameters, since it is expected that the effects of noise are very much suppressed by the long inspiral of those waveforms. If then, the shifts are recovered at around 5σ, this would strengthen the explanation given earlier for the smaller shifts being (partly) due to noise. Furthermore, this test could be extended to include different shift values to find the sensitivity threshold of each. Additionally, non-GR waveforms that are not simply a shift in one of the deviation parameters could be included. This test could even be used to find at what level a specific alternative theory of GR can be detected by the parameterised test. However, that method is designed to be a generic test of GR, as illustrated with the results of this section. Not the exact shifts in the injected waveforms are measured, but the consistency of the data with the GR model.

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Conclusions In this work, several aspects of the reliability of the parameterised test of GR have been investigated. Firstly, the accuracy of the waveform model IMRPhenomPv2 has been studied by comparing it with the model SEOBNRv2 which has been independently calibrated against numerical relativity (NR) waveforms. Overlaps between the models have been calculated around the parameters for GW150914 and for GW151226 (only with non-precessing spins for the latter). It was found that overlaps were high enough such that inaccuracy in the model should not be detectable beyond the limit set by noise, except for high spins (χ1 > 0.5 in the worst case, but for most configurations overlaps were high up to much higher spin values). Lower overlaps could mostly be explained by a lack of NR calibration waveforms used in the models, especially with one high positive and one high negative spin, and high spins in general for non-equal mass configurations. Lastly, it was found that overlaps are generally worse for stronger precession effects. Secondly, the robustness of the parameterised test against fluctuations in the detector noise was studied. It is expected that offsets of 2–2.5 σ in the inspiral coefficients measured for GW150914 are due to noise. The parameterised test was run on fifteen different noise realisations with an injected NR waveform, yielding a maximum offset of 2.07σ. However, most runs found a significantly higher SNR than that of GW150914, such that lower offsets are expected and the results of the test deemed inconclusive. Thirdly, the sensitivity of the test for GW150914 was probed by running it on IMRPhenomPv2 model waveforms with a built-in deviation from GR injected in a stretch of data. Not all deviations were recovered as significantly as injected, possibly due to noise for the inspiral parameters. Repeating the investigation for a model waveform like GW151226 can provide evidence for or against this explanation. The shifts in parameters from the intermediate and merger-ringdown regime were recovered with much greater significance for the negative shifts than for the positive. This could be due to the structure of the shifts in the models, were a negative shift around one yields a much larger fractional change than a positive one. It is possible that a different, unknown effect changes the sensitivity of the test depending on the size and shape of the deviation.

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Appendix A

Results for the noise investigation

Figure A.1: Results for the parameterised test on a numerical relativity waveform injected in different stretches of noise (see Section 4.1). The figure continues on the following pages, including noise 1–15 (except noise 10 for which no runs are finished at the time of writing). The posteriors are given as violin plots, with bars indicating the 90% confidence interval. Coefficients from the inspiral regime of the model IMRPhenomPv2 are indicated by their post-Newtonian order. The last panel contains the recovered coefficients from the intermediate and merger-ringdown regimes. Some of the results are not finished yet, which is where there are bars instead of violin plots

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Appendix B

Results for the non-GR waveforms

Figure B.1: Results of the parameterised test on model waveforms (IMRPhenomPv2) with shifted inspiral parameters (see Section 4.2). For the top row, δ χ ˆ3 was shifted with +0.4 (blue plots) and -0.4 (green plots) in the injected waveform. For the bottom row, δ χ ˆ4 was shifted with +3.3 (blue plots) and -3.3 (green plots). The posteriors are given as violin plots, with bars indicating the 90% confidence interval. Coefficients from the inspiral regime of the model IMRPhenomPv2 are indicated by their post-Newtonian order. The last panel contains the recovered coefficients from the intermediate and merger-ringdown regimes.

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Figure B.2: Results of the parameterised test on model waveforms (IMRPhenomPv2) with shifted intermediate or merger-ringdown parameters. For the top row, δ βˆ2 was shifted with +0.7 (blue plots) and -0.7 (green plots) in the injected waveform. For the second, third and last row, this was δ βˆ3 = ±0.8, δ α ˆ 2 = ±1.3 and δ α ˆ 4 = ±1.6, respectively (with blue plots for the positive and green plots for the negative shifts). The posteriors are given as violin plots, with bars indicating the 90% confidence interval. Coefficients from the inspiral regime of the model IMRPhenomPv2 are indicated by their post-Newtonian order. The last panel contains the recovered coefficients from the intermediate and merger-ringdown regimes. Some of the results are not finished yet, which is where there are bars instead of violin plots. 56