The Pennsylvania State University The Graduate School Eberly College of Science
SEARCHING FOR GRAVITATIONAL WAVES FROM NEUTRON
STARS
A Dissertation in Physics by Ashikuzzaman Idrisy
© 2015 Ashikuzzaman Idrisy
Submitted in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
May 2015 The dissertation of Ashikuzzaman Idrisy was reviewed and approved∗ by the following:
Benjamin J. Owen Professor of Physics Dissertation Advisor, Chair of Committee
Martin Bojowald Professor of Physics
Michael Eracleous Professor of Astronomy
Sarah Shandera Assistant Professor of Physics
Nitin Samarth Professor of Physics George A. and Margaret M. Downsbrough Department Head
∗Signatures are on file in the Graduate School.
ii Abstract
In this dissertation we discuss gravitational waves (GWs) and their neutron star (NS) sources. We begin with a general discussion of the motivation for searching for GWs and the indirect experimental evidence of their existence. Then we discuss the various mechanisms through which NS can emit GWs, paying special attention the r-mode oscillations. Finally we end with discussion of GW detection. In Chapter 2 we describe research into the frequencies of r-mode oscillations. Knowing these frequencies can be useful for guiding and interpreting gravita- tional wave and electromagnetic observations. The frequencies of slowly rotating, barotropic, and non-magnetic Newtonian stars are well known, but subject to various corrections. After making simple estimates of the relative strengths of these corrections we conclude that relativistic corrections are the most important. For this reason we extend the formalism of K. H. Lockitch, J. L. Friedman, and N. Andersson [Phys. Rev. D 68, 124010 (2003)], who consider relativistic polytropes, to the case of realistic equations of state. This formulation results in perturbation equations which are solved using a spectral method. We find that for realistic equations of state the r-mode frequency ranges from 1.39–1.57 times the spin frequency of the star when the relativistic compactness parameter (M/R) is varied over the astrophysically motivated interval 0.110–0.310. Following a successful r-mode detection our results can help constrain the high density equation of state. In Chapter 3 we present a technical introduction to the data analysis tools used in GW searches. Starting from the plane-wave solutions derived in Chapter 1 we develop the F-statistic used in the matched filtering technique. This technique relies on coherently integrating the GW detector’s data stream with a theoretically modeled wave signal. The statistic is used to test the null hypothesis that the data contains no signal. In this chapter we also discuss how to generate the parameter space of a GW search so as to cover the largest physical range of parameters, while keeping the search computationally feasible. Finally we discuss the time- domain solar system barycentered resampling algorithm as a way to improve to the computational cost of the analysis.
iii In Chapter 4 we discuss a search for GWs from two supernova remnants, G65.7 and G330.2. The searches were conducted using data from the 6th science run of the LIGO detectors. Since the searches were modeled on the Cassiopeia A search paper, Abadie et. al. [Astrophys. J. 722,1504–1513, 2010], we also used the frequency and the first and second derivatives of the frequency as the parameter space of the search. There are two main differences from the previous search. The first is the use of the resampling algorithm, which sped up the calculation of the F-statistic by a factor of 3 and thus allowed for longer stretches of data to be coherently integrated. Being able to integrate more data meant that we could beat the indirect limit on GWs expected from these sources. We used a 51 day integration time for G65.7 and 24 days for G330.2. The second difference is that the analysis pipeline is now more automated. This allows for a more efficient data analysis process. We did not find a credible source of GWs and so we placed upper limits on the gravitational wave strain, ellipticity, and r-mode amplitude of the sources. The best upper-limit for the strain was 3.0 × 10−25, for ellipticity it was 7.0 × 10−6 and for r-mode amplitude it was 2.2 × 10−4.
iv Table of Contents
List of Figures viii
List of Tables ix
Acknowledgments x
Chapter 1 Introduction 1 1.1 Indirect evidence ...... 2 1.2 Theory of gravitational waves ...... 4 1.2.1 Linearized gravity ...... 6 1.2.2 Generation of gravitational waves ...... 8 1.3 Neutron stars ...... 12 1.3.1 Gravitational waves from neutron stars ...... 14 1.3.2 r-modes ...... 15 1.4 Detection of gravitational waves ...... 17 1.4.1 History and Development ...... 17 1.4.2 Operating Principles ...... 18 1.4.3 Data Analysis challenge ...... 20 1.4.4 Detectors of the future ...... 22
Chapter 2 R-mode frequencies for slowly rotating neutron stars with real- istic equations of state 23 2.1 Introduction ...... 23 2.1.1 General relativity ...... 26 2.1.2 Rapid rotation ...... 27 2.1.3 The crust ...... 28 2.1.4 Other effects ...... 28 2.1.5 Summary ...... 29
v 2.2 Formulation ...... 29 2.2.1 Equilibrium Solution for non-rotating star ...... 30 2.2.2 Interpolation Schemes ...... 31 2.2.3 Equilibrium solution for slowly rotating star ...... 35 2.2.4 Perturbation equations ...... 36 2.2.4.1 Perturbations of non-rotating stars ...... 36 2.2.4.2 Perturbations of slowly rotating stars ...... 37 2.2.5 Boundary Conditions ...... 39 2.3 Numerical Solution ...... 41 2.3.1 Chebyshev-Galerkin Method ...... 41 2.3.2 Finding κ ...... 43 2.3.3 The eigenfunctions ...... 45 2.4 Results ...... 46 2.4.1 The n = 1 polytrope ...... 46 2.4.2 Tabulated Equations of State ...... 47 2.5 Discussion ...... 51 2.6 Conclusion ...... 54
Chapter 3 Fundamentals of gravitational wave data analysis 56 3.1 Introduction ...... 56 3.2 Signal model ...... 57 3.3 Maximum likelihood detection ...... 60 3.4 Template spacing ...... 63 3.4.1 Sphere-covering ...... 63 3.4.2 Template Spacing Metric ...... 64 3.4.3 Lattice generation ...... 66 3.5 Resampling ...... 67 3.5.1 Implementation ...... 70 3.5.2 Speed up ...... 71 3.5.3 Issues ...... 72
Chapter 4 Using barycentric resampling in gravitational wave searches from two supernova remnants 74 4.1 Introduction ...... 74 4.2 Data analysis method ...... 77 4.2.1 Data Prepration ...... 83 4.2.2 Search ...... 84 4.2.3 Validation ...... 85
vi 4.2.4 Upper Limits ...... 88 4.3 Implementation ...... 88 4.4 Results ...... 90 4.5 Discussion ...... 92
Chapter 5 Conclusion 98
Bibliography 100
vii List of Figures
1.1 Energy loss from Hulse-Taylor binary system ...... 3 1.2 Regions of influence of gravitational object ...... 5 1.3 Polarization of gravitational waves ...... 8 1.4 Digram of gravitational wave source ...... 10 1.5 Visualization of r-mode oscillatios ...... 16 1.6 Schematic of LIGO detectors ...... 18 1.7 GW passing a LIGO interferometer ...... 19 1.8 LVC interferometer sensitivity ...... 20
2.1 Convergence of eigenvalue ...... 45 2.2 Plot of eigenfunctions ...... 46 2.3 Plot of eigenfunctions ...... 47 2.4 Eigenvalues for n=1 polytrope ...... 48 2.5 Eigenvalues for realistic equations of state ...... 49 2.6 Compare eigenvalue for different equations of state ...... 50
3.1 Schematic of resampling ...... 68
4.1 First 12 steps of the pipeline ...... 78 4.2 Steps 13 to 17 of the pipeline ...... 79 4.3 Veto bands ...... 86 4.4 2F distribution ...... 87 4.5 Schematic of DAG ...... 90 4.6 Strain amplitude upper limit for G65.7 ...... 92 4.7 Ellipticity upper limit for G65.7 ...... 93 4.8 r-mode upper limit for G65.7 ...... 94 4.9 Strain amplitude upper limit for 330.2 ...... 94 4.10 Ellipticity upper limit for G330.2 ...... 95 4.11 r-mode upper limit for G330.2 ...... 96
viii List of Tables
2.1 We present a list of all the tabulated EoS, for each EoS we show the stable maximum mass, the radius for a 1.4M star, κ for a compactness of .15, the coefficients for the quadratic fit of the κ 2 M M of the form a + b R + c R , and the root mean square error (RMSE) of the EoS data points to the quadratic fit...... 51 2.2 This table gives the numerical κ for all tabulated EoS, over the range of compactness values considered in our analysis. These values are plotted in Fig. 2.5. The “-" indicates a compactness that could not be obtained with that EoS...... 55
4.1 Search targets and astronomical parameters ...... 83 4.2 Target objects, and astronomical parameters used in each search along with information about the SFTs used in the searches. . . . . 83
ix Acknowledgments
First and foremost I would like to thank my research advisor Professor Benjamin Owen, for none of this would be possible without his support. Thank you for taking a chance on a untested second year graduate student. Thank you for teaching me about the stars. Thank you for molding me into a scientist. Thank you for giving me the freedom to explore different career opportunities. Thank you for all the guidance given, when real-life came knocking. Thank you, so very much, for everything. I would like to thank professors Bojowald, Eracleous, and Shandera for being on my committee. Thank you for taking the time to read and edit this thesis and for overseeing my comprehensive examination. I would like to thank Dr. Ra Inta without whom I could not have finished this work. Thank you for taking the time to help me debug code, for teaching me about Bash, vim, Perl, and Python, and for all the practical advice. I would like to thank Dr. Ian Jones who I had the pleasure of collaborating with for the material presented in Chapter 2 of this thesis. This work really wouldn’t have been the same without his input. I would like to thank my parents for teaching me to love school. If it wasn’t for this I may have not ventured all the way to graduate school. I also want to thank my brother for always being a constant source of fun and interesting conversation. My physics degree has helped him quite a bit with his own education over the years, and for that I am happy. Finally I would like to thank all the friends I’ve made in State College during my time here. They have all helped a tremendous amount in both my professional and personal development. I hope the universe finds a way for us to stay close.
x Dedication
To my wife and son ...
Amanda you’ve been there since the beginning of my graduate career, and your personal sacrifices and encouragement has made it all possible. Thank you for always believing in me, even when I didn’t believe in myself . . . Our son, although a new addition to the team, has provided me with plenty of motivation to put all the pieces together and finish this degree. I thank you both so, so much.
xi Chapter 1 | Introduction
Gravitational waves (GWs) were theoretically predicted by Albert Einstein in 1915 as a part of his theory of General Relativity (GR). They are ripples in space-time generated by various astrophysical phenomena. A major scientific goal of the last century has been to observe these waves directly. To meet this challenge extremely sensitive detectors have been built by the Laser Interferometer Gravitational Wave Observatory (LIGO) and Virgo collaboration, otherwise known as the LVC. LIGO operates two 4 km long interferometers, one at at Livingston, LA and one at Hanford, WA. Virgo operates a 3 km interferometer near Pisa, Italy. These detectors have recently been upgraded, and beginning next year data will be collected by the advanced LIGO (aLIGO) and advanced Virgo (aVirgo) detectors. These detectors will be capable of detecting instantaneous strain amplitudes on the order of 10−25 [1], an unprecedented level of precision in any scientific experiment. GW detectors will allow us to see further into the cosmos than ever before. This is because GWs are not as susceptible to scattering due to the interstellar medium that affects astronomical observations, since gravity interacts weakly with matter. This does not mean that astronomical observations are to be replaced. Instead they will be even more in demand, since a detection of a GW will be followed up by astronomical observations (whenever possible). In fact, GW observations and astronomical observations are quite complementary. Not only can an astronomical follow up verify a GW detection but a GW signal can alert astronomers of new ob- jects to study. The two types of observations also carry complementary information about an object. GWs carry information about macroscopic properties, whereas the astronomical observations carry information about the microscopic physics at play. The idea of taking multiple observations of a single object is at the heart of
1 multi-messenger projects, of which LIGO would be one measurement device, see [2] for a review. A flurry of research has already been done in anticipation of the new detectors, and new projects are currently being done by members of the LVC in preparation for the advanced detectors. The two projects discussed in this thesis are both related to the scientific goals of the LVC. The research discussed in Ch. 2 asks: at what frequencies will gravitational waves be observed from the r-mode oscillations of neutron stars? The results presented in Ch. 2 can be used by both (electromagnetic) astronomical and GW observatories. In Ch. 4 we describe searches for GWs from two supernova remnants which have not been previously examined. We also describe work done to enhance the analysis pipeline which is used for directed GW searches. In Sec. 1.1 of this chapter we discuss measurements that provide indirect evidence for the existence of gravitational waves. In Sec. 1.2 we present the theoretical description of GWs including their generation, propagation, and energy carried. In Sec. 1.3 we discuss neutron stars and how they act as sources of gravitational wave radiation. Finally, in Sec. 1.4 we present details on the LIGO detectors, including history, operating principles, data analysis techniques, and future work.
1.1 Indirect evidence
The first indirect measurement of GWs was presented in reference [3]. When studying the orbital decay rate of binary pulsar system PSR B1913+16, the authors discovered that the observed decay rate was not consistent with conservation of energy, unless one accounted for the loss of energy through gravitational radiation. In a recent follow up of the same binary, reference [4] has shown that the results of [3] are still true today. In Fig. 1.1 we see that over thirty-five years of data on this binary system show that the decay rate follows the predictions of general relativity. A more contemporary experiment carried out by the BICEP2 [5] collaboration seemed to provide more indirect evidence of GW. The goal of this experiment was to search for primordial B-mode polarization of the cosmic microwave background (CMB) radiation [5]. A B-mode is twisting polarization pattern generated by gravitational waves from inflation that get imprinted on the CMB. Initially, the results of the BICEP2 experiment seemed to indicate a B-mode detection. However
2 Figure 1.1: This figure shows how the observed orbital decay of the Hulse-Taylor binary system PSR B1913+16 (black diamonds) matches the decay rate due to gravitational waves theoretically predicted by General Relativity (the solid line). It is clear that the data match very closely with the theory for this data stretch of 35 years, which indicates the existence of gravitational waves. Figure from reference [4]. a more recent analysis conducted by reference [6] shows that the data analyzed by the BICEP2 experiment was contaminated by dust fields in the local galaxy. It is possible that future experiments like BICEP2 will yield promising results. Even though there is some indirect evidence of their existence, there are several
3 reasons why a direct observation of GWs is important. The first is that it would further verify the validity of Einstein’s theory of General Relativity, and provide data which can be used to test alternative theories of gravity. Second, direct detections will allow us to probe the structure of neutron stars, and ascertain properties of black holes. Indeed, the detections of GWs is not the end of the story but rather just the beginning. Now that we have the motivation to search for GWs let us examine how they are generated.
1.2 Theory of gravitational waves
Gravitational wave solutions are only one part of a much larger picture of the gravitational field of an object. In general what we experience as the gravitational field is really the curvature of the space-time generated by mass and energy. This space-time curvature is described by the mathematics of manifolds on which we define a distance via the metric tensor, gαβ. The metric tensor is a solution of Einstein’s equation, 1 R − g R = 8πT (1.1) αβ 2 αβ αβ where Rαβ is the Ricci tensor, R the Ricci scalar, and Tαβ is the stress-energy tensor. The left hand side of Eq. 1.1 is a description of the space-time curvature in and around the object, and the right hand side describes the material properties of the object. In order to better understand GWs it will be important to understand the gravitational field (space-time curvature) all around the object. Due to the difficulty of solving these equations various approximations will be applied. Let us start with Fig. 1.2 below. The source of gravitation has some characteristic length, L, and mass, M. We will assume that the constituents of the source are moving slowly compared to the speed of light. This means that the wavelength of the GWs, λ, is much larger than L, or M (in geometrized units of length). We will also assume that outside of the dashed (dark green) circle the gravitational field is described the by metric hαβ, and that this metric is a weak perturbation of the flat-spacetime metric, which means | hαβ| 1. These approximations allows us to label sections at distances away from the source according to the physical approximation valid at those distances. For
4 Linearized Gravity
λr
Weak Field Near Zone
Local Wave Zone
Distant Wave Zone
Figure 1.2: This figure shows the various zones of influence of the gravitational object (in green) in the center of the concentric circles. Inside the dashed (dark green) region full non-linear GR must be applied. Beyond this region out to the dotted (green) circle, linearized gravity can be applied. The field solutions inside the Local Wave Zone are wave-like solutions which can be traced back to static solutions inside of the Weak Field Near Zone. distances less than r ≈ 30M, the gravitational field is dominated by the object and one might have to solve the full non-linear Einstein equation, possibly through numerical simulations, see [7–9]. Beyond this zone we enter the Weak Field Near Zone (WFNZ), where the gravitational field can be assumed to be static. The field in this region can be extended into the next region, the Local Wave Zone (LWZ) using linearized gravity. Finally in the Distant Wave Zone, the gravitational field of other objects, as well as the space-time curvature, must be taken into account
5 and we treat the propagation of GWs with geometric optics. By making the weak-field and slow-motion assumptions we can use linearized gravity to describe the field of the object between the distances marked by the dark green and light green circle. We will use the plane-wave solutions that arise in linearized gravity to extend the solutions from the LWZ back into the WFNZ.
1.2.1 Linearized gravity
In linearized gravity we decompose the metric tensor into two parts. The first part is given by a metric that has zero or small amounts of curvature. For simplicity we take the flat Minkowski metric, ηαβ, which has zero curvature. The other part is the weak field hαβ mentioned above, which can be treated as a perturbation of the flat metric. The full metric is then
gαβ = ηαβ + hαβ. (1.2)
We will see that hαβ has plane-wave solutions, and hence they are suitable for representing the gravitational wave field. In order to find these solutions we start by assuming that the waves are observed far-away from their source, then we can substitute gαβ into the vacuum Einstein equation
1 R − g R = 0 (1.3) αβ 2 αβ which gives
1 (∂ ∂ hσ + ∂ ∂ hσ − ∂ ∂ h − h − η ∂ ∂ hρλ + η h) = 0, (1.4) 2 σ β α σ α β α β αβ αβ ρ λ αβ where is the flat-space d’Alembertian operator. Some simplifications must be made in order to solve this equation. First, we introduce the trace-reversed perturbation 1 h¯ = h − h (1.5) αβ αβ 2 αβ where h = ηαβh . Then the Einstein equation is
¯µ ¯µ ¯ ¯ ∂α∂µhβ + ∂µ∂βhα − ηµν∂µ∂νhαβ − ηαβ∂µ∂νhµν = 0 (1.6)
6 Next, we use the freedom to choose the coordinates and employ the commonly used Lorenz gauge ¯ ∂αhαβ = 0 . (1.7)
Picking the Lorenz gauge means seeking a coordinate transformation x0 = x + ξ, where ξ is a solution of ¯µ ξβ = ∂µhβ. (1.8)
There is a further freedom in the Lorenz gauge because any solution of ξβ = 0 can be added to ξβ and still be a solution. In this gauge the Einstein equations reduce to ¯ hαβ = 0. (1.9) This equation represents a wave equation for the trace-reversed metric perturbation, which has the solution ¯ i[~k·~x−ωt] hαβ = aαβ e . (1.10) ~ Where k is the wave vector, aαβ is a symmetric 4×4 matrix, and ω = kc is the wave frequency. Plugging Eq. 1.10 into the wave equation shows that the wave vector satisfies the condition α k kα = 0 (1.11) which means kα is a null vector and thus GWs propagate at the speed of light. Substituting Eq. 1.10 into the Lorenz gauge condition (Eq. 1.7) gives
β i[~k·~x−ωt] ik aαβ e = 0. (1.12)
β β Which means k aαβ = 0, and since k represents the direction of propagation this says that GWs propagate transverse to their direction of travel. If we take positive z as the direction of propagation, and use the further gauge freedom of the Lorenz 1 gauge to pick ξ, aαβ can be made traceless, this is known as the transverse traceless
1We do not put the bar over the perturbed metric in the transverse traceless gauge since trace reversing it is meaningless.
7 Figure 1.3: We see the effect of a gravitational wave on a ring of test masses. The deformations at the top are those solely due to waves with “+” polarizations, and at the bottom those due to “x” polarizations. This figure is from [10].
(TT) gauge. Putting this all together we get the solution for the perturbed metric,
0 0 0 0 TT 0 h+ h× 0 i[kz−ωt] h = e . (1.13) αβ 0 h× −h+ 0 0 0 0 0
It should be noted that hαβ in Eq. 1.4 started with 10 degrees of freedom (dof) but after picking the coordinates, i.e. picking the Lorenz gauge, that was reduced to 6 dof. After a ξ was selected, i.e. employing the further freedom in the Lorenz
gauge, there were 2 dof remaining. The two amplitudes components, h+ and h× are those 2 degrees of freedom, and physically they represent the two possible polarizations of GWs. The naming comes from the effect each polarization has on a ring of test masses, see Fig. 1.3. The deformations at the top of Fig. 1.3 are those solely due to waves with “+” polarizations, and at the bottom those due to “x” polarizations.
1.2.2 Generation of gravitational waves
As mentioned above we will show how the wave solutions found from linearized gravity which propagate in the LWZ are generated. To accomplish this we follow
8 the method of multipole expansion (similar to that found in electrodynamics) for gravitational waves (see [11]) to find solutions of Eq. 1.1. As a consequence of the slow-motion assumption time-derivatives are un-important in the WFNZ because the source is moving quasi-statically. This means we only have to solve the three dimensional version of Eqs. 1.1 and 1.7, which are
¯0j ¯jk h,j = 0 h,k = 0 (1.14) ∇2h¯00 = 16T 00 = 16ρ (1.15) ∇2h¯0j = 16ρvj (1.16) ∇2h¯jk ∼ O(v/c)2 (1.17) where ρ represents the mass density of the object, and ρvj represents the momentum density. We will begin with Eq. 1.15 and discuss Eq. 1.16 towards the end of this section, and ignore the higher order Eq. 1.17 altogether. We see that Eq. 1.15 is essentially the Poisson equation for a Newtonian potential, given we map h00 to the Newtonian potential Φ on the LHS of the equation, and divide by a factor of 4 on the RHS of the equation. Therefore, the solution for h00 is the same as it would be for Φ given a mass distribution 4ρ, namely
Z ρ(x0) h00 = 4Φ = 4 dx0 , (1.18) |x − x0| where x is the field point, and x0 is a point inside the source, see Fig. 1.4. In this figure r is defined as the distance from the field point to the center of mass (COM), r0 is the distance from the source point to the COM, and n is a unit vector pointing form the COM to the field point x. In order to compute this integral we turn to Taylor expansion. First rewrite the denominator in the equation above as
1 1 1 = q = √ (1.19) |x − x0| (x − x0)2 r2 − 2n · x’ + r02 then pull out a factor of 1/r and expand using r0 r to get
j j k 1 02 j k 1 1 1 njx 3 njnkx x − 3 r δjkn n q = + + + ... (1.20) r 2n·x0 r0 2 r r2 2 r3 1 − r2 + ( r )
9 Source
r n x COM r′
|x-x'| x′
Figure 1.4: Digram of gravitational wave source
Then the multipole solution for h00 is
"1 Z nj Z 3 njnk Z r0 # h¯00 = 4 ρ(x0)d3x0 + ρ(x0)x03d3x0 + ρ(x0)[x x − ]d3x0 + ... r r2 2 r3 j k 3 (1.21) The first integral term in 1.21 is the mass monopole, the second is the mass dipole, 02 and the third term Ijk = xjxk − r /3, is the mass quadrupole. We would like to extend this solution out to the LWZ. In this zone time derivatives can no longer be ignored. In keeping with the electrodynamics analogy we want to find the terms in Eq. 1.21 which will oscillate and there by give off radiation in the LWZ. The first term in Eq. 1.21 is the mass monopole, and due to conservation of mass, there will be no oscillations from this term. The next term, the mass dipole, is a measure of the linear momentum; and due to conservation of linear momentum there will be no oscillations from this term either. Therefore the first term in the multipole expansion that can to contribute to GWs is the mass quadrupole (Ijk). In the LWZ linearized gravity can be applied, and so we again have Eq. 1.9, which we already know has wave solutions. Given the fact that the first contributing term to GWs is Ijk, we can make the ansatz for a wave solution of the type ¯00 h ∼ 2(1/r)Ijk(t − r). This is not the correct solution because the LHS has no indices and the RHS has two free indices. To fix this we take derivatives with
10 respect to the spatial coordinates xj and xk on the RHS, so the solution is
1 h i h¯00 = 2 Ijk(t − r) . (1.22) r ,jk
To find the other components of h¯αβ we can use the Lorenz gauge conditions. Given ¯00 ¯0j h,0 = −h,j we get 1 h¯0j = −2 I˙jk(t − r) , (1.23) r ,k ¯jk ¯j0 and using h,k = h,0 gives 2 h i h¯ = I¨ (t − r) . (1.24) jk r jk We can again employ the transverse traceless gauge to get the full solution
2 h iTT hTT = I¨ (t − r) . (1.25) jk r jk
This last equation shows that the most dominant contributor to GW in the transverse-traceless gauge comes from the 2nd time derivative of the mass quadrupole moment. The energy carried away as radiation (or the GW luminosity) from this term is
dE ω6 = I 2 (1.26) dt 16π zz where ω is the rotation rate of the source, and
I − I = xx yy (1.27) Izz is the equatorial ellipticity and the z-axis is chosen as the rotation axis of the star, and Ijk is the moment of inertia tensor. Another contributor to GW radiation is the second time derivative of the momentum density. The expansion for these current multipole moments follows the one presented above, but the starting place is the mass current term (ρvj) in Eq. 1.16. The radiation resulting in the LWZ is
8 1 TT hTT = − naS¨ (t − r) (1.28) jk 3 r jab bk
11 where Z a c 3 Sbk = (bacx ρv )xkd x. (1.29)
In most situations, like mergers, supernovas, and binary sources (see Sec. 1.3 below) the mass quadrupole moment dominates the GW emission because the current v multipole moment has an extra factor of ( c ). However in special cases like the r-mode oscillations for neutron stars, see Sec. 1.3.2 below, the current multipole moment is more dominant. In general we can write the contributions from all the multipoles as expansions in spherical harmonics as,
∞ +l !l 1 X X d l,m E2,l,m l,m B2,l,m hjk = I Tjk + S Tjk (1.30) r l=2 m=−l dt where
v u E2,l,m u2(l − 2)! T = t r2∇ ∇ Y l,m − trace (1.31) jk (l + 2)! j k B2,l,m E2,l,m Tjk = NqTq(j k)pq (1.32)
In Eq. 1.31 the “trace” means to only keep the trace part, and Y l,m is the spherical harmonic function. We can also write the energy radiated (GW luminosity) in terms of the multipole expansion as,
* !l+1 2 !l+1 2+ dE 1 ∞ +l d d X X l,m l,m = I + S (1.33) dt 32π l=2 m=−l dt dt
Where the angled brackets represent time-averaging.
1.3 Neutron stars
We will now discuss neutron stars (NS), which are one of the most promising sources of GWs for the detectors used by the LVC. Neutron stars represent one final end point in the stellar evolution. Stellar evolution beings with a cloud of gas and dust in the interstellar medium. If there is a perturbation of a cloud at its Jeans mass [12] the cloud will collapse
12 towards its center of mass. During this collapse the gravitational energy is converted to heat via compression, and temperature of the core reaches approximately 107 K, at which point the hydrogen gas begins to burn and helium is formed [13]. As the star ages, heavier elements are burned which provides a outward pressure against the inward force of gravity and this keeps the star in hydrostatic equilibrium. When the fuel for nuclear burning is exhausted, gravity becomes the more dominant force and the star begins to collapse in on itself.
If the star has an initial mass of 2–5 M [14], the nuclear burning proceeds from hydrogen to helium burning. This helium burning will result in a carbon, and then oxygen core. If the progenitor star is closer to 5 M then it is possible that carbon, and then oxygen burning will occur. When the star is no longer capable of nuclear burning the C-O core will start to collapse. For stars in this mass range the core collapse is halted by the electron degeneracy pressure and the star becomes a white dwarf (WD).
If the mass of the progenitor star ranges from approximately 5–60 M [14], the burning process continues from carbon, to oxygen, to silicon. As the silcion burns it creates an iron core. The iron core cannot burn to form heavier elements to maintain hydrostatic equilibrium and the core collapses. In this case the core collapse is stopped by neutron degeneracy pressure and the star will become a NS.
If the progenitor has a mass greater than 60 M , neutron degeneracy pressure is not large enough to counteract the gravitational force, and the star forms a black hole [14]. In the case of NS, the iron-rich core which remains after the collapse becomes a mixture of superfluid neutrons, superconducting protons, and relativistic degenerate electrons, through the process of inverse beta decay and neutronization [12]. Usually, the collapse will lead to a supernova event. That is, the collapse will happen very quickly and result in a huge release of energy. The existence of NS was hypothesized by Baade and Zwicky based on observations of supernovas [15]. Only two years after the discovery of the neutron by Chadwick in 1932 [16]. It is possible that the remnants left over after a supernova will contain a NS and this is why we search for GW from supernova remnants (SNRs).
All of the NS discovered to date are ≈ 1–2 M and have radii between 10–14 km [17, 18]. The prototypical NS has a mass of 1.4 M and a radius of 10 km. The average density of such a star is, ρ = 6.65 × 1014 g cm−3, which is roughly
13 14 −3 3 times nuclear density, ρnuc = 2.3 × 10 g cm . Temperatures of newly born NS are O(1012K) [19], a minute or so after the supernova event, and after about 1000 years they cool to O(106K) [12]. They exhibit large magnetic fields on the order of, O(1014G), due to the conservation of magnetic flux. Neutron stars also exhibit large rotation rates because angular momentum must be conserved during the supernova, and so a large slowly rotating body gets transformed into a small rapidly rotating one. For all that is known about NS one key piece of the puzzle is still missing. This is the nuclear equation of state (EoS), the functional relationship between the pressure and energy density. Due to the fact that kBT and other energy scales are small compared to degeneracy pressure for NS, it is believe that there is one equation that should describe all NS. Since this ‘canonical’ EoS has not been discovered, a common procedure in research is to consider the star as having a polytropic equation of state. Another approach taken by some researchers is to use tabulated EoS. These tables have columns of pressure, energy density, number density, etc. that are filled in from a theoretical model of the star. In Chapter 2 we present the first calculations of the r-mode oscillation frequency which use these tabulated EoS.
1.3.1 Gravitational waves from neutron stars
There are several ways that NS can produce GWs. If the NS is in a binary system with another NS or a BH, the binary will emit GWs due to its changing quadrupole moment as its components orbit. As the binary gets closer to merging and the orbit shrinks faster due to GW emission it becomes one of the so-called compact binary coalescence (CBCs) sources, searched for by GW detectors. The binary orbit can be divided into an inspiral, merger, and ring down phase. An upper limit on the gravitational wave strain from these signals is approximately 1.7 × 10−23 [20]. Another potential source of GWs comes from NS which are created from supernovas. This and other similar catastrophic events, like gamma ray bursts, will create so-called burst signals in the GW detectors. The upper limit on the gravitational wave strain from these signals is approximately 6.0 × 10−21 [20]. The different NS sources can create an incoherent background signal. This stochastic signal is the GW analog of the cosmic microwave background radiation.
14 An upper limit on the gravitational wave strain from these signals is approximately 5.6 × 10−22 [20]. Another source of GWs is due to NS with non-axisymmetric deformations, such as crustal deformations (mountains), or cracks in the crust [21], or the star being accreted upon non-isotropically [22]. These sources give rise to continuous wave signals, named as such because the GW signal from these sources can last as long as or longer than the observation time, and remain essentially monochromatic. An upper limit on the gravitational wave strain from these signals is approximately 2.2 × 10−24 [20].
1.3.2 r-modes
Individual NS can also radiate GWs through oscillation modes. This falls under the continuous waves category mentioned above. There are various stellar oscillations that can occur in NS. They are the p-modes, which are restored by the pressure in the star, the g-modes which are restored by the buoyant force, the f -modes which are fundamental p-modes, and r-modes which are restored by the Coriolis force. Under certain conditions these modes can become unstable to gravitational wave radiation. The instability can be understood as follows. If one takes a pulsation mode, that in the stars reference frame travels in a direction opposite to the star’s rotation, then that mode has negative angular momentum [23]. And, if the star is rotating fast enough that it can drag the mode along with its rotation, then in an inertial frame the mode appears to be rotating in the same direction as the star, and the mode emits positive angular momentum. This radiation lowers the total angular momentum of the star as seen in the inertial frame, but in the star’s reference frame it makes the angular momentum more negative. These more negative modes will then radiate more GW in the inertial frame. It was realized early on that this instability would be difficult to observe because for most modes (even in perfect fluid stars where there is no viscosity to compete with the instability) the stellar rotation rate would have to be near the Keplerian velocity of the star, i.e. the star would need to rotate so fast that it would start breaking apart. It was shown by [24,25] that r-mode oscillations are unstable to gravitational wave radiation for any rate of rotation. It was later shown by [26] that the r-mode instability could be detected by LIGO even with some realistic
15 (a) (b)
Figure 1.5: Visualization of r-mode oscillatios viscosity. Whereas other sources of continuous waves come from the mass quadrupole moment of the source, Eq. (1.25), GWs from r-mode oscillations come from the current multipole moment, Eq. (1.28), because these modes represent the motion of fluid currents at constant radial distances, see Fig. 1.5. To see why this is so we consider r-modes for Newtonian slowly rotating stars. In this case we can write down the analytical form for the mode perturbation,
r l δv = αΩR Y B,l,leiωt (1.34) j R j where α is a dimensionless constant, Ω is the angular rotation rate of the star, R is the radius of the star, and ω is the oscillation frequency. The magnetic-parity B,l,m vector spherical harmonic, Yj is
B,l,m 1 k p l,m Yj = q jkprN ∇ Y (1.35) l(l + 1) where N k points out from the center of the star. The perturbation velocity field defined in Eq. 1.34 is orthogonal to the radial direction and is given shape by Y l,m. The perturbations make ellipses on the constant radial surfaces of the star, a visualization which is shown in Fig. 1.5. To first order the frequency is
(l − 1)(l + 2) ω = − Ω + O(Ω3). (1.36) l + 1
This shows that the r-mode frequency is proportional to the rotation rate of the star, as it should be since these modes are restored by the Coriolis force. There
16 has been a lot of research aimed at finding the r-mode frequencies to higher orders within the context of General Relativity. The main goal of the next chapter is to find the r-mode frequencies for fully relativistic stars with realistic (tabulated) equations of state.
1.4 Detection of gravitational waves
1.4.1 History and Development
Having discussed the sources of GW signals we move onto a discussion of the detectors designed to find these signals. The original gravitational wave detectors were large aluminum resonant bars created by Dr. Joseph Weber at University of Maryland [1, 20] in 1965. The idea was to make the physical dimensions of the bar such that they would oscillate when GW passed the Earth, then resonance would cause the bar to deform and a measurement of this deformation would be taken. Resonant bars are still in use today, for example: Auriga, Nautilus, and MiniGRAIL, see [1] for more details. They can achieve instantaneous strain sensitivities, h0(t), of approximately 10−21 [1]. Although this is quite a high sensitivity to achieve, the interferometeric method used by the LVC can achieve greater sensitivity, of approximately 10−23 [1] see Fig. 1.8 below. Furthermore, interferometers are better suited for the task of finding GW because they can achieve greater sensitivity for a broad range of frequencies, whereas the bar mass detectors are only effective in narrow bands near the resonance of the bar. The LIGO detectors2 were built by the California Institute of Technology and Massachusetts Institute of Technology, in the early 1990s. The phases of development are known as Initial LIGO from 2000–2007, enhanced LIGO from 2009–2013, and advanced LIGO from 2015 onwards. During these phases there were fixed periods in between upgrades and maintenance when data were collected. These data collection times are known as science runs and the detectors are said to be in science mode. Thus far there have been 6 science runs, S1-S6. S1-S4 happened during the Initial LIGO phase. S5 occurred during the initial LIGO phase, and was the longest science run with data taken from November 2005 to
2We focus on the LIGO detectors because the data analyzed in Ch. 4 was obtained from these detectors.
17 Figure 1.6: Schematic description of LIGO detectors, from Spacetime and Geometry: An Introduction to General Relativity [27].
October 2007. S6 occurred during the enhanced LIGO phase. The data analyzed in this work comes from S6, with data collected from July, 7, 2009 to Oct, 21, 2010.
1.4.2 Operating Principles
The LIGO detectors are Fabry-Perot interferometers, but we can illustrate them more simply as Michelson interferometers, see Fig. 1.6. In this design, laser light is sent to a beam splitter where it travels down the two arms of the interferometer, each of length L. The light is bounced back and forth between the mirrors approximately 100 times before escaping the beamsplitter and heading to the photodiode. If a gravitational wave were to pass through the interferometer it would perturb the relative positions of the test masses (the mirrors) by an amount δL and the strain,
δL ∼ h (1.37) L represents the magnitude of the gravitational wave h. The interferometer is set up so that if there is no change in the positions in the mirrors (no gravitational wave) then the two beams of light will destructively interfere at the photo-diode.
18 Figure 1.7: This graphic shows what happens when a GW passes by LIGO interfer- ometer. The blue rectangles represent the mirrors and the red lines represent the laser light. The green half-circle is the photodiode. This figure is from [28].
In Fig. 1.7 we show a cartoon version of what happens when a GW passes by an interferometer. In reality the interferometers are far more advanced than just Michelson interferometers, making use of Fabry-Perot cavities, power-recycling mirrors, etc., more details can be found in [1,28,29]. As mentioned in the previous section an optimistic magnitude of an instanta- neous gravitational wave strain is 10−21. Due to the sensitivity required to find such a faint signal one must be very aware of the sources of noise present in these detectors. Some of the common sources of noise and systematic uncertainty in the data come from the suspension cables for the test masses and the power lines. Two sources of noise that are truly limiting are the photon back-reaction on the mirror and seismic noise. Figure 1.8 shows the sensitivity in strain for the 4 km detectors at Hanford (H1) and Livingston (L1), and for the 3 km detector in Pisa (V1). The shot noise is seen as the steep rise in the noise at the RHS of the Fig. 1.8. This noise comes from the uncertainty in the phase of the laser light due to the quantization of light. The seismic noise is seen as the steep rise in the LHS of Fig. 1.8. This noise comes from vibrations of the ground create by winds, ocean waves, atmospheric
19 Figure 1.8: This figure shows the sensitivity in strain for the 4 km detectors at Hanford (H1), Livingston (L1), and the 3 km detector in Pisa (V1). The shot noise is seen as the steep rise in the noise at the RHS of the figure. The seismic noise is seen as the steep rise in the LHS of the figure. disturbances, heavy traffic, etc. To combat the noise from the mechanical and electrical instruments much of the equipment is run at a frequencies much lower or much higher outside of the sensitivity range set by the photon-shot, and seismic noise.
1.4.3 Data Analysis challenge
From the noise profile in Fig. 1.8 we can see that continuous wave signals with amplitude of order 10−24 will be buried by the noise. In order to extract these signals specialized data analysis techniques must be used [30]. Fortunately for us this is not an entirely new problem and has been an issue since the early days of radar [31]. One solution to finding weak signals buried in noise is a technique called
20 matched filtering. This method relies on having an accurate model for the signal one is trying to find. Then one coherently integrates the model signal h(t) with the detector’s data stream x(t),
2 Z Tspan/2 (x||h) = x(t)h(t)dt (1.38) Sh(f0) Tspan/2 where Sh(f0) is the one-sided power spectrum of the detector evaluated at expected frequency of the signal f0, and Tspan is the length of data that is integrated. From standard statistics we know that the likelihood ratio is the probability that the data stream contains both a signal and noise to the probability that it only contains noise. The likelihood ratio is used to construct the F-statistic,
B(x||h )2 + A(x||h )2 − 2C(x||h )(x||h ) 2F = 1 2 1 2 D B(x||h )2 + A(x||h )2 − 2C(x||h )(x||h ) + 3 4 3 4 . (1.39) D where
A ≈ 2(h1||h1) ≈ 2(h3||h4),B ≈ 2(h2||h2) ≈ 2(h4||h4) 2 C ≈ 2(h1||h2) ≈ 2(h3||h4),D = AB − C . (1.40)
and h1, h2, h3, h4 represent different components of the modeled signal. In Ch. 3 we provide a more technical discussion of the F-statistic as well as the models used to describe the continuous wave signal. Furthermore we discuss issues, like how many filters (model signals) need to the used and how to properly space them to completely cover the parameter space of the signals modeled, without exceeding the computational resources available. In Chapter 4 we present a search for GW from two known supernova remnants (SNRs), which makes use of the technical material discussed in Ch. 3. In Ch. 4 we also discuss the physical properties of the SNRs and the search method (pipeline) used to find to GW. Having found no GW we use the pipeline to place upper limits on the gravitational strain, the ellipticity and the r-mode oscillation amplitudes from the neutron stars in these SNRs.
21 1.4.4 Detectors of the future
One of the main goals in the design of future detectors is to reduce the noise. The aLIGO detectors will make use of a more powerful laser, and larger mirrors, in order to combat the photon shot noise. After the aLIGO detectors are decommissioned new upgrades will take place and we will have the LIGO III detectors [32, 33]. The KAGRA detector which is planned for 2020 in Japan will be underground to reduce the noise from gravity gradient and will also cryogenically cool the mirrors to combat photon shot noise [34]. There are also plans to build a detector in India. In the not-so-distant future we will see the advent of space based detectors, with the LISA pathfinder mission paving the way with its launch next year [35]
22 Chapter 2 | R-mode frequencies for slowly rotating neutron stars with re- alistic equations of state
2.1 Introduction
In this chapter we explore the r-mode frequencies of neutron stars and discuss how they can be useful in guiding and interpreting gravitational wave and electromagnetic observations. R-modes [36] are non-radial stellar oscillations which can become unstable to gravitational wave (GW) emission via the CFS [37,38] mechanism [24,25], even in the presence of viscosity [39,40]. This makes r-modes a promising source of gravitational waves for ground based detectors [22, 26, 41]. The energy thus radiated has been used to explain the spins of newly born neutron stars [39, 40] and of accreting neutron stars [22,41]. The r-modes have also been proposed as a model for quasi-periodic oscillations of low mass X-ray binaries, and for burst oscillations of accretion-powered millisecond X-ray pulsars (AMXPs) [42]. Possible detections of r-modes in X-ray oscillations have been made from AMXPs XTE J1814−338 [43] and 4U 1636−536 [44]. (It has been argued [45] that the discovery in [43] is inconsistent with the spin-down of the pulsar, though an alternative explanation has been offered by reference [46].) Because of its physical importance many authors have calculated the r-mode frequency. Results are often given in terms of the rotating frame mode angular
23 frequency σR, or in dimensionless form as σ κ ≡ R , (2.1) Ω where Ω is the rotational angular velocity of the star. Reference [36] showed that r-modes are rotationally restored oscillations and so their frequencies are proportional to the stellar rotation frequency. The authors calculated that for slowly and uniformly rotating Newtonian stars, κ is equal to a constant which is independent of the equation of state (EoS):
2m κ = κ = , (2.2) 0 l(l + 1) where l and m are spherical harmonic indices. It was shown by [47] that for barotropic stars the r-modes must satisfy l = |m|. The l = m = 2 r-mode, for which κ0 = 2/3, is the most susceptible to the CFS instability [39,40]. Reference [48] extended the slow-rotation expansion for Newtonian stars and found corrections for κ to second order in the rotation rate of the star. Reference [49] examined slowly rotating relativistic stars to leading order in the rotation rate and accounted for metric perturbations. References [50–52] examined rapidly rotating relativistic stars using the Cowling approximation, in which metric perturbations are neglected. The rotational and relativistic corrections to κ are dependent upon the EoS used to model the star. The studies mentioned thus far used polytropic EoS to simplify the calculation of κ. In this paper we present the first calculation of κ for stellar models constructed from realistic (tabulated) EoS. We use a subset of the EoS studied by [53,54]. EoS which can not support a maximum mass of least 1.85 solar masses are excluded from our analysis. This mass is a conservative upper limit derived from the 99.7% confidence limit of the observed “1.97 M ”pulsar, see Fig. 2 of [18] for more details. This left 14 EoS for which a range of κ values were calculated over a range of masses. There are several applications of our results. Our calculation of κ can be used to interpret electromagnetic observations of r-modes, such as those (possibly) of [43, 44]. Our calculation can also be used in collaborative work between GW detectors and (electromagnetic) astronomical observatories. Assuming r-mode
24 GWs are detected from a previously unknown pulsar, our range of κ would give electromagnetic astronomers a frequency band in which to search for pulsations from rotation. If one has the r-mode frequency (from GW data) and the pulsation frequency (from electromagnetic data), it is possible to get the pulsar’s compactness which might be used to constrain the EoS. Finally, for GW searches conducted on pulsars with known spin frequencies, such as the Crab, our results define a narrow frequency band over which to search for GWs from r-modes. A narrow band search [55] has already been conducted on the Crab but it was not looking for gravitational waves from r-modes. Rather the search was centered around the usual, two times the spin-frequency of the pulsar. Our results can also be used by a new narrow-band search pipeline [56] which claims to be twice as sensitive as the previous Crab search. The outline of the paper is as follows. In Sec. 2.1 we estimate how general relativity affects the r-mode frequency in comparison to other physical phenomena such as the star’s crust, rotation rate, magnetic fields, and stratification. In Sec. 2.2 we present the formulation of the r-mode oscillation problem found in [49]. In Sec. 2.3 we discuss the numerical methods used to solve the equations that arise from this formulation. We also give more details than [49], including convergence details for our code, and accuracy estimates for our results. In Sec. 2.4 we discuss the results of applying our numerical solution to both polytropic and realistic equations of state, with a focus on the latter. Finally in Sec. 2.5 we draw conclusions from our results and examine the aforementioned applications in light of these results. Throughout we use geometrized units, where Newton’s gravitational constant and the speed of light are unity. We now estimate the importance of various corrections to the Newtonian slow- rotation r-mode frequency. In common with much of the literature, we make use of the dimensionless rotating frame mode frequency κ defined in Eq. (2.1). Note that the corresponding gravitational wave emission will be at the inertial frame mode frequency σI, related to the rotating frame mode frequency by:
σI = (κ − m)Ω = σR − mΩ. (2.3)
For the l = m = 2 r-mode of a slowly rotating Newtonian star, σR = 2Ω/3 while
|σI| = 4Ω/3. Strictly, σI = −4Ω/3; the negative sign is a consequence of the
25 opposite sense of rotation of the patterns produced by the mode as viewed in the inertial and rotating frames. This opposite sense of rotation is responsible for the CFS instability. It follows that a decrease of κ by a small fraction will increase the inertial frame gravitational wave frequency by half of that fraction. This must be borne in mind when designing gravitational wave searches (see Sec. 2.5 for some discussion of detection issues).
2.1.1 General relativity
The importance of general relativistic effects can be estimated by looking at the ratio of stellar mass to radius, M/R, a dimensionless measure of the compactness of the star: M M ! 10 km! ≈ 0.207 . (2.4) R 1.4M R
It follows that departures from the Newtonian κ = κ0 results, at the level of a few tens of percent, can be expected when relativistic effects are included. This expectation is confirmed by the post-Newtonian and fully relativistic numerical calculations of [49], who considered polytropic stars. For instance, for a uniform density star with M/R = 0.207, they found that κ/κ0 ≈ 0.85, a reduction of ∼ 15% compared to the Newtonian case. Note that (consistent with the results to be presented in this paper), [49] found that the effect of relativity is to decrease the rotating frame mode frequency, and therefore increase the inertial frame frequency. Given that we will conclude that relativistic effects are likely to be the dominant factor influencing the r-mode frequency, it is worth considering how large a spread of the compactness parameter might be found in the neutron star population. This is of immediate astrophysical interest, as it would determine how large a range in gravitational wave frequency must be searched when looking for r-mode emission from a pulsar of known spin frequency (but unknown compactness). In the numerical calculations that follow, for each chosen realistic EoS, a range of masses from 1 M up to a value close to the maximum mass (specific to that equation of state) are considered. The lower limit of 1 M was adopted since the lepton-rich, hot matter in supernova explosions most likely does not support proto-neutron stars with smaller masses. Also, most measured masses with tight error bars are greater than this; see Fig. 1 of [17]. Thus we have taken the conservative lower limit of 1 M .
26 This led to a range of compactnesses 0.110 ≤ M/R ≤ 0.310. The graphs of r-mode frequency that follow are plotted over this range. That this is a sensible estimate of the range of possible compactnesses of realistic neutron stars can be confirmed from Fig. 2 of [57], which shows mass-radius curves for a large collection of realistic EoS, some of which are also considered in this paper. As illustrated, there is a hard upper limit of M/R . 0.350 that comes from the constraint that the EoS be causal. The maximum mass members of soft EoS come close to (but do not quite reach) this limit, e.g. the EoS AP4, which is one of the EoS considered here. In terms of a lower limit on compactness, low mass stars with stiff equations of state are relevant. Reference [57] find EoS whose 1 M members have R ≈ 14.5 km, corresponding to M/R ≈ 0.103. However, for the EoS considered here this lower limit would result in stars with masses less than 1 M . Therefore we increase this lower limit to 0.110. Taken together, we see that the range of compactnesses considered in this paper includes neutron stars presently considered realistic in the literature. We will return to this in Sec. 2.5, after having calculated the range in r-mode frequencies corresponding to this range in compactness.
2.1.2 Rapid rotation
The effect of stellar rotation on the r-mode frequency was considered by [48], who calculated the leading order correction to the mode frequency, as quantified by a parameter κ2 satisfying Ω2 κ = κ0 + κ2 . (2.5) πGρ¯0
Here ρ¯0 is the average mass density of the corresponding non-rotating star and κ2 is dimensionless, of order unity, and dependent upon the equation-of-state. The factor 2 !2 3 Ω fspin R 1.4M = 0.145 6 (2.6) πGρ¯0 716 Hz 10 cm M is a dimensionless measure of the effect of rotation on the star. We have scaled the spin frequency fspin = Ω/2π to a value of 716 Hz, the spin rate of the fastest observed millisecond pulsar [58]. Taking a representative value of κ2 ≈ 0.29 from [48], we see that rapid rotation can increase the value of κ by ∼ 6% for the fastest rotating stars, while rotational corrections rapidly become negligible for more slowly spinning stars. It follows that rotational effects can indeed be significant, but probably never
27 dominate the relativistic ones. Note that the sign of the frequency shift corresponds to a decrease in the gravitational wave frequency, and so acts oppositely to the relativistic effects described above.
2.1.3 The crust
The presence of a solid crust is very important for r-mode damping [59,60], and it can have an effect on the mode frequency as well. In addition to Coriolis restoring forces acting throughout the star, there are also elastic restoring forces in the crust. Information on how this influences the r-mode frequency can be extracted from Fig. 1 of [60]. For sufficiently slow rotation rates, the mode frequency is close to the standard κ = κ0 = 2/3 result, with the fluid core but not the solid crust participating in the motion. For sufficiently high rotation rates the mode frequency is again close to κ = κ0 = 2/3, but now the whole star, crust plus core, participates in the motion. For intermediate spin rates, there is an avoided crossing, which means that the ‘r-mode’ is more accurately described as a hybrid rotational–elastic mode.
From Fig. 1 of [60] it seems that the departure from the κ = κ0 = 2/3 result is significant (i.e. more than a few percent and can be discerned by eye) over the spin frequency interval 0.05 . Ω/ΩK . 0.1, where ΩK is the Keplerian angular velocity of the star. Being dependent upon the EoS, this quantity is not known accurately, but taking a representative value of ΩK/(2π) ∼ 1500 Hz, this corresponds to the spin interval 75 < fspin/Hz < 150, so crustal corrections could be relevant for some milli-second pulsars. Looking at the right hand panel of Fig. 1 of [60], we see that departures from
κ = κ0 = 2/3 of ∼ ±20% are possible. This is comparable with the shift of Sec. 2.1.1, but is double-sided, i.e. the mode frequency might be shifted up or down. However, the modification of the mode frequency at this level only applies over a narrow range in spin frequency so it is unlikely to affect most stars.
2.1.4 Other effects
There will be other factors that will have effects on the r-mode frequency. We very briefly mention two more here. Real neutron stars are stratified, with radial entropy and composition gradients.
28 The effect of stratification was considered by [61], who found that while the majority of the inertial modes are significantly affected by stratification, the nodeless l = m r-modes are relatively unaffected; see Fig. 4 of [61]. The effect of stratification on inertial modes was also investigated by [62], who found the r-mode frequency was affected only very slightly. This was shown to be true even for very rapidly rotating stars; see Fig. 12 of [62]. Magnetic fields will also alter the r-mode frequency, but the effect is again likely to be slight, see [63, 64], or the numerical simulations of [65]. Physically, the smallness of the corrections corresponds to the magnetic restoring forces being small compared the Coriolis restoring forces. It should, however, be pointed out that the above references consider non-superconducting stars. The effect of superconductivity may make magnetic corrections more important, but quantitative estimates of such effects are not currently available.
2.1.5 Summary
We have presented estimates of the importance of various effects on the r-mode spin frequency, using some simple estimates and results from the literature. General relativistic effects can have a significant influence on the r-mode frequency, at the level of tens of percent. The effects of rapid rotation are insignificant in all but the fastest spinning pulsars. The effects of an elastic crust are slightly more difficult to quantity, but it seems likely they will only be competitive with relativistic ones in rather narrow intervals in stellar spin frequency, and so are unlikely to be significant in the majority of the known pulsars. The likely dominance of relativistic effects motivates the careful treatment of relativistic stars with realistic EoS presented in the remainder of this paper.
2.2 Formulation
In this section we summarize the formulation of the r-mode oscillation problem in order to provide context, and establish terminology and notation. The full details are available in [49,66]. In this formulation, the PDEs resulting from the perturbed Einstein equations for a perfect fluid star, are turned into ODEs via spherical harmonic expansion. This expansion is only possible if the star is assumed to be
29 slowly rotating. In order to derive the perturbation equations, equilibrium solutions of slowly rotating and non-rotating stars must be found first.
2.2.1 Equilibrium Solution for non-rotating star
The non-rotating equilibrium solution is found by solving the Einstein equations
Gαβ = 8πTαβ, where Gαβ is derived from the line element:
ds2 = −e2ν(r)dt2 + e2λ(r)dr2 + r2dθ2 + r2sin2θdϕ2, (2.7)
Tαβ is the energy-momentum tensor for a perfect fluid:
Tαβ = ( + p)uαuβ + pgαβ, (2.8)
(r) is fluid energy density, p(r) is fluid pressure, and
uα = e−νtα (2.9)
α α is the fluid 4-velocity with t = (∂t) the time-like Killing vector. Applying this information and comparing Gαβ = 8πTαβ term-by-term leads to the Oppenheimer- Volkov (OV) equations [27]. Therefore solving the OV equations is equivalent to solving the Einstein equations. The OV equations must be solved numerically for most equations of state. The numerical solution is better realized when one uses the enthalpy, h, of the star instead of the radial distance, r, as the dependent variable [67] (we refer to these as the OVL equations). The OVL equations for a non-rotating star are
dr r(r − 2M) = − , (2.10) dh (M + 4πr3p) and dM dr = 4πr2 , (2.11) dh dh where M(h = 0) is the mass of the star. The metric functions λ and ν are found using
ν(h) − νc = hc − h, (2.12)
30 where 1 2M(h = 0)! ν = −h + ln 1 + (2.13) c c 2 r(h = 0) and 1 " 2M(h)# λ(h) = − log 1 − . (2.14) 2 r(h) Just like the OV equations the OVL equations are singular at the center of the star. Therefore, the numerical integration is started near the center, h = hc, using the following truncated power series solutions
1 " # 2 ( 3(hc − h) 1 3 r(h) = 1 − c − 3pc + 1 2π(c + pc) 4 5 (h − h) ) × c , (2.15) (c + 3pc)
4π 3 31 M(h) = cr (h) 1 + (hc − h) , (2.16) 3 5c where c is the central energy density, pc is the central pressure and
d = − . (2.17) 1 dh h=hc
The integration is carried out to h = 0, which is guaranteed to be the surface of the star.
2.2.2 Interpolation Schemes
In order to solve the OVL equations one also needs to specify an equation of state. First we discuss the simple case of polytropic EoS, which is given by
n+1 p = Kρ n (2.18) = ρ + np (2.19) where n is the polytropic index, ρ is the rest-mass-energy density and K is a constant. In geometrized units K has dimensions (length)2/n. It is better for numerical calculations to work in dimensionless units so the following transformation
31 is made
p → K−np ρ → K−nρ → K−n.
The EoS is then
n+1 p = ρ n (2.20) = ρ + np (2.21)
If we define the co-moving enthaply
Z p dp0 h(p) = (2.22) 0 (p0) + p0
For polytropic stars this means
h 1 i h(p) = ln 1 + (n + 1)p n+1 (2.23) or
eh − 1!n ρ(h) = (2.24) n + 1 eh − 1!n+1 p(h) = (2.25) n + 1 eh − 1!n " eh − 1!# (h) = 1 + n . (2.26) n + 1 n + 1
Plugging these into Eqs. (2.10)-(2.43) with the initial values
!n " !# ehc − 1 ehc − 1 = 1 + n (2.27) c n + 1 n + 1 !n+1 ehc − 1 p = (2.28) c n + 1 !n ne2hc ehc − 1 1 = − . (2.29) (ehc − 1) n + 1
32 gives us the OVL equations for polytropic EoS. Realistic EoS are presented as tables with columns given by values of pressure, pi, energy density, i, and baryon number density ni, where the i subscript indexes the row of the table. The values in the columns must be interpolated in order to get a well-behaved EoS which can be used with an OV solver. We use the interpolation scheme of [54] for our analysis. The interpolation scheme of [54] assumes a power law relationship between pressure and energy density, p ci+1 = , (2.30) pi i where log(pi+1/pi) ci+1 = , (2.31) log(i+1/i) for i ≤ ≤ i+1. Using this and the definition of the co-moving enthalpy Eq. 2.22 a column of values, hi = h(pi), is generated and is used to get the piecewise function
(h) =
( " # )1/(ci+1−1) i + pi ci+1 − 1 i i exp (h − hi) − (2.32) pi ci+1 pi (2.33)
for hi ≤ h ≤ hi+1. Using Eq. (2.30) one can get p(h). The interpolation scheme of [68] by contrast assumes a power law relationship between pressure and the number density
n γi p(n) = pi , (2.34) ni where ln pi+1 − ln pi γi = , (2.35) ln ni+1 − ln ni for ni ≤ n ≤ ni+1. This scheme requires creating an auxiliary column of values ˜i,
! 1 pi+1 pi ˜i+1 = ˜i + − (2.36) γi − 1 ni+1 ni
33 to get the auxiliary energy density ˜(n),
1 p pi ˜(n) = ˜i + − (2.37) γi − 1 n ni which is used to find the energy density
(n) = n [˜(n) + mn] , (2.38)
where mn is the mass of a neutron. For our purposes we require (h), and p(h).
Thus we use Eq. (2.22) to find a column of values for hi. Then we interpolate the hi and ni values using cubic splines, to define the function n(h), which we substitute into (n), and p(n). Finally, in our simple spline interpolation (see [69] for details) scheme we assume a power law relationship between pressure and energy density. Whereas the interpolation schemes of [54, 68] take the first law of thermodynamics into account this alternative spline scheme does not. Therefore this scheme mainly serves as a test of how much the first law affects the result for κ.
We used a linear interpolation of log pi and log i values to determine the power law between the points. Making use of Eq. (2.22) to get values for hi, we interpolated hi with pi and i using a quadratic spline, to get the function p(h) and (h) respectively. The quadratic spline was used because first order splines lead to discontinuities in the solutions to the OVL equations, and the third order splines lead to extra inflection points in the solutions. To check that our results are robust to the interpolation method used, we compared the scheme in [54] to that found in [68], and our own spline interpolation for a sample group of EoS. We found that the percent difference in κ from the different schemes was less than 0.3% overall, and in some cases less than 0.1%. From these small percent differences, we see that our code is robust to the interpolation scheme used.
34 2.2.3 Equilibrium solution for slowly rotating star
The equilibrium solution for a rotating star is again found by solving the Einstein equations. This time, Gαβ is derived from the line element:
ds2 = −e2ν(r)dt2 + e2λ(r)dr2 + r2dθ2 (2.39) + r2sin2θdϕ2 − 2ω(r)r2sin2θdtdϕ with the definition ω¯(r) ≡ Ω − ω, (2.40) where ω(r) accounts for the the frame-dragging effect. The line element in Eq. (2.39) is only correct up to order Ω. This slow rotation limit means that the star retains its spherical geometry, since the centrifugal deformation of its figure is an order Ω2 effect [70]. In the rotating case the fluid 4-velocity becomes
uα = e−ν(tα + Ωϕα) (2.41)
α α α α where t = (∂t) and ϕ = (∂ϕ) , are respectively the time-like and rotational Killing vectors. For a rotating star the equilibrium solution comes from solving the OVL Eqs. (2.10)–(2.11) and the Hartle equation [70]. In the enthalpy formulation, which was extended to slow rotation by [53], the Hartle equation is broken up into a pair of first order ODEs as
dω¯ dr = e(ν−νc+λ)f (2.42) dh dh and df 4 dr = 16π( + p)e−(ν−νc−λ)ω¯ − f . (2.43) dh r dh These equations are also singular at the center of the star, and so the following power series are used to begin the integration,
( ) 12(c + pc) ω¯(h) = ωc 1 + (hc − h) , (2.44) 5(c + 3pc)
( " # ) 16π 5 6(2c − 3pc) 1 f(h) = (c + pc)ωcr(h) 1 + + (hc − h) (2.45) 5 7 5(c + 3pc) (c + pc)
35 After solving Eqs. (2.10)–(2.11) , and Eqs. (2.42)–(2.43) we numerically invert the solution for Eq. (2.10). Using this we change the functions λ(h), ν(h),... into λ(r), ν(r), etc.
2.2.4 Perturbation equations
In this section we sketch out the derivation of the perturbation equations, full details of which are in [49]. The metric and fluid perturbation terms in these m m m equations are expanded in terms of scalar (Yl ), vector (r∇Yl , r × ∇Yl ) and m tensor (∇βYl uγ) spherical harmonics. This basis makes it possible to classify perturbations as axial or polar parity. Axial parity modes have the same parity m m m as r × ∇Yl , whereas polar parity modes have the parity of Yl and ∇Yl . The generic oscillation will be a combination of polar and axial modes. However, the leading order term in the expansion will either be of polar parity or axial parity. This leads to the terminology of axial-led and polar-led modes, with r-modes being the former. This classification works for rotating as well as non-rotating stars because the parity of the leading order term does not change due to rotation.
2.2.4.1 Perturbations of non-rotating stars
The equilibrium configuration for a non-rotating star is given as a solution to the OVL Eqs. (2.10)–(2.11). The perturbations are
m δ = δ(r)Yl , (2.46) m δp = δp(r)Yl , (2.47) (1 1 ) δuα = H (r)Y mtα + W (r)Y mrα + V (r)∇αY m P 2 0 l r l l ×e−ν, (2.48) α (λ−ν) αβγδ m δuA = −U(r)e ∇βYl uγ∇δ r. (2.49)
α α Notice that δuP is of polar parity and δuA is of axial parity. Employing the Regge- Wheeler gauge and expanding the metric in tensor spherical harmonics, the metric
36 perturbation of polar-parity mode can be written
2ν H0(r)e H1(r) 0 0 2λ P H1(r) H2(r)e 0 0 m h = Y , (2.50) µν 2 l 0 0 r K(r) 0 0 0 0 r2 sin2 θK(r)
and that of an axial-parity mode can be written
m m 0 0 −h0(r) csc θ∂ϕYl h0(r) sin θ∂θYl m m A 0 0 −h1(r) csc θ∂ϕYl h1(r) sin θ∂θYl h = (2.51) µν symm symm 0 0 symm symm 0 0
where “symm” indicates components obtained by symmetry.
The final step in finding the perturbation equations is to examine δGαβ =
8πδTαβ term-by-term. This leads to 10 differential equations for (H0,H1,H2, K, h0, W, V, U, δ, δp).
The equations decouple into equations for (H1, h0, W, V, U) and (H0,H2, K, δ, δp). Under the assumption of linear stability, reference [71] showed that for non-radial
oscillations H0 = H2 = K = δ = δp = 0. Thus the perturbation equations of O(1) in Ω are −(ν+λ) h ν+λ i0 Vl[l(l + 1)( + p)] − e ( + p)e rWl = 0, (2.52) " # 2 00 2 0 0 0 2 2 2λ 0 0 0 0 r hl −r (ν +λ )hl + (2−l −l)r e −r(ν +λ )−2 hl −4r(ν +λ )Ul = 0. (2.53)
In the Newtonian limit Eq. (2.52) corresponds to conservation of mass and Eq. (2.53), which relates the metric perturbation to the fluid perturbation, reduces to identity (vanishing metric perturbation). We have made slight algebraic changes to Eq. (2.52) and Eq. (2.53) from the way they appear in [49], so that they are easier to use in the numerical computation discussed in Sec. 2.3.
2.2.4.2 Perturbations of slowly rotating stars
Similar to the non-rotating case the fluid perturbation is decomposed into spherical scalar and vector harmonics. But this time the Lagrangian perturbation formalism is used. In general the Lagrangian change of a quantity Q, ∆Q, is related to the
37 Eulerian change, δQ, via ∆Q = δQ + LξQ, where Lξ is the Lie derivative with respect to the fluid displacement vector ξ. Here, the displacement vector is defined as:
∞ ( ) α 1 X 1 m α α m α µβγδ m iσt ξ ≡ Wl(r)Yl r + Vl(r)∇ Yl − iUl(r)P µ ∇βYl ∇γ t∇δ r e , iκΩ l=m r (2.54) where α (ν+λ) α α P µ ≡ e δ µ − tµ∇ t . (2.55)
It should be noted that Wl,Vl are of polar parity whereas Ul is of axial parity. Again using the Regge-Wheeler gauge the metric perturbation is
2ν m m m m m H0,l(r)e Yl H1,l(r)Yl h0,l(r)( )Yl ih0,l(r) sin θ∂θYl sin θ ∞ m 2λ m m m m X H1,l(r)Yl H2,l(r)e Yl h1,l(r)( )Yl ih1,l(r) sin θ∂θYl iσt h = sin θ e , µν 2 m l=m symm symm r Kl(r)Yl 0 2 2 m symm symm 0 r sin θKl(r)Yl (2.56) where the polar parity components are H0,l,H1,l,K, and the axial parity components are h0,l, h1,1. The coefficients can be grouped as:
Wl,Vl,Ul,H1,l, h0,l ∼ O(1), (2.57)
H0,l,H2,l,Klh1,l, δ, δp ∼ O(Ω). (2.58)
The O(1) coefficients obey the O(1) Eqs. (2.52)–(2.53). The definition of ξα leads to κΩ terms in the perturbation equations, and so only the O(1) variables are kept in the perturbation equations. This ensures that the order of the equations is no higher than O(Ω). Reference [66] derives the O(Ω) equations by invoking the conservation of circulation for an isentropic fluid, which gives
" # e2ν [l(l + 1)κΩ(h + U ) − 2mωU¯ ] + (l + 1)Q ∂ r2ωe¯ −2ν W − 2(l − 1)¯ωV l l l l r r l−1 l−1 " # e2ν − lQ ∂ r2ωe¯ −2ν W + 2(l + 2)¯ωV = 0, (2.59) l+1 r r l+1 l+1
38 " # (l − 1) (l − 2)Q Q −2∂ ωe¯ −2νU + ∂ r2ωe¯ −2ν U l−1 l r l−2 r2 r l−2
" −2ν −2ν +Ql (l − 1)κΩ∂r (e Vl−1) − 2m∂r (¯ωe Vl−1)
# m(l−1) 2 −2ν −2ν 16πr(+p) 1 2λ + r2 ∂r (r ωe¯ ) Vl−1 + (l − 1)κΩe (l−1)l − r e Wl−1
" −2ν −2ν 2 2 + mκΩ∂r [e (hl + Ul)] + 2∂r (¯ωe Ul) (l + 1)Ql − lQl+1
# 1 2 −2ν h 2 2 2 i + r2 ∂r (r ωe¯ ) Ul m + l(l + 1) Ql+1 + Ql − 1
" −2ν −2ν −Ql+1 (l + 2)κΩ∂r (e Vl+1) + 2m∂r (¯ωe Vl+1)
# m(l+2) 2 −2ν −2ν 16πr(+p) 1 2λ + r2 ∂r (r ωe¯ ) Vl+1 + (l + 2)κΩe (l+1)(l+2) − r e Wl+1
" # (l + 2) +(l + 3)Q Q 2∂ ωe¯ −2νU + ∂ r2ωe¯ −2ν U = 0, (2.60) l+1 l+2 r l+2 r2 r l+2 where the constants Ql are defined as
" (l + m)(l − m) #1/2 Q ≡ . (2.61) l (2l − 1)(2l + 1)
2.2.5 Boundary Conditions
In order to solve Eqs. (2.52)–(2.53) and Eqs. (2.59)–(2.60) we need to apply the appropriate boundary conditions. Notice that the perturbation equations are a set of linear ODEs. This indicates that multiplying a solution by a constant gives another solution, which means that the boundary conditions must take the form ζ(hl,Ul,Vl,Wl) = 0, where ζ represents an arbitrary linear combination. Alternatively, the boundary condition must be given in terms of a condition on a 0 logarithmic derivative, e.g. Ul /Ul = constant. First let us consider the boundary conditions near the center of the star.
39 These are also known as the regularity conditions:
r l r l+1 U (r → 0) = U (r),W (r → 0) = W (r) l R l l R l r l r l+1 h (r → 0) = h (r),V (r → 0) = V (r) l R l l R l (2.62) where R is the surface of the star, and the barred functions are slowly varying. Only two of these boundary conditions are linearly independent as shown by [49]. Next let us examine the boundary conditions at the surface of the star. The Lagrangian perturbation of the pressure is zero at the surface, which leads to
Wl(R) = 0. (2.63)
Note that hl is the only unknown function defined outside of the star, where it obeys 2M d2h "l(l + 1) 4M # 1 − l − − h = 0, (2.64) r dr2 r2 r3 l which has the exact solution
∞ l+s X ˆ R hl(r) = hl,s , (2.65) s=0 r and (l + s − 2)!(l + s + 1)!(2l + 1)! 2M s hˆ = hˆ . (2.66) l,s s!(l − 2)!(l + 1)!(2l + s + 1)! R l,0
The sum in Eq. (2.65) is the hypergeometric function 2F1 (l − 1, l + 2; 2l + 2; 2M/r), ˆ see [72]. The factor hl,0 is arbitrary, as it corresponds to an overall normalization ˆ of the perturbation. assuming hl,0 = 1.
Matching the interior and exterior solutions for hl(r) completes the boundary conditions. The first matches the function at the surface,
lim [hl(R − ε) − hl(R + ε)] = 0. (2.67) ε→0
The second, which is given by the condition on the Wronskian, matches the
40 derivatives at the surface,
0 0 lim [hl(R − ε)h (R + ε) − h (R − ε)hl(R + ε)] = 0. (2.68) ε→0 l l
Both conditions must be true for all values of l.
2.3 Numerical Solution
Following the formulation presented in [49], as summarized by Sec. 2.2 , has effectively changed the problem of finding κ from solving the dynamical Einstein equations to solving coupled ODEs for spherical harmonic expansion coefficients. Solving the perturbation Eqs. (2.52)–(2.53), and (2.59)–(2.60), with the boundary conditions Eqs. (2.62)–(2.63) and (2.67)–(2.68) is analytically intractable, but numerically feasible. The first step is to solve the OVL equations and find the equilibrium functions λ(r), ν(r), ω(r), p(r) and (r). Next, insert the regularity conditions explicitly into the perturbation equations. Since the eigenfunctions, hl,Ul,Vl+1,Wl+1, represent coefficients of an infinite series we must truncate at maximum value for l, let us denote it as lmax, to get a finite number of equations. Because we are focusing on axial-led hybrid modes we need to set lmax to be a odd number in order to get a closed system of equations. This choice of axial-led hybrids also means we solve for the eigenfunctions, hl,Ul,Wl+1,Vl+1, where l = m, m + 2 ..., and set the others to zero. Next, note that each term in the perturbation equations and boundary conditions can be written as the product of a background (equilibrium) function
B(r) and foreground (perturbation) function Fl(r). For example, in the term ¯ l 4r(λ + ν)Ul(r) the background function is 4r(λ + ν)(r/R) and the foreground function is Ul.
2.3.1 Chebyshev-Galerkin Method
We solve the perturbation equations by expanding both B(r) and Fl(r) in Chebyshev polynomials. Chebyshev polynomials have the form
Ti(y) = cos(i arccos y), i = 1, 2, 3,... (2.69)
41 and are defined on the domain [−1, 1]. For our purposes the Chebyshev polynomials’ most important property is their exponential convergence when approximating well-behaved functions [69]. In general one can express any well-behaved function S(y) on the domain [−1, 1] in terms of these polynomials as
i Xmax 1 S(y) = siTi(y) − s0, (2.70) i=0 2 where imax represents the highest order Chebyshev polynomial that is used to approximate the function. The coefficients si are extracted using the following formula: imax+1 " 1 !# 1 ! 2 X π(j + 2 ) πi(j + 2 ) si = S cos cos (2.71) imax j=0 imax imax In order to make use of these functions we have to change the domain of our functions from [0,R] to [−1, 1] using
r y = 2 − 1. (2.72) R
Thus we transform B(r) into B(y) and expand as,
i Xmax 1 B(y) = bi Ti(y) − b0, (2.73) i=0 2 similarly for Fl(r) we have,
i Xmax 1 Fl(y) = fl,i Ti(y) − fl,0. (2.74) i=0 2
Let us define the derivative of Fl(r) as
imax 0 d X 0 1 0 Fl (y) = Fl = fl,i Ti(y) − fl,0. (2.75) dr i=0 2
0 Here the prime notation in fl,i does not mean derivative, rather it is a numeric 00 coefficient for the derivative expansion of a function. Similarly we define Fl (y) for 00 second derivatives of Fl(r), and fl,i as its coefficients. With these definitions in place, the terms in the perturbation equations will
42 0 00 be of the form B(y)F (y), or B(y)Fl (y), or B(y)Fl (y). There is a relationship between the f 0 and f coefficients given by the identity: li li
0 0 fl,i − fl,i+2 = 4(i + 1)fl,i+1. (2.76)
It should be noted that Eq.(2.76) has a 4 whereas the standard formula, e.g. in [69], has a 2. We include the extra factor of 2 to transform d/dy to d/dr via Eq.(2.72). The identity Eq.(2.76) can be used twice to find the relationship between f 00 and li f . Since f 0 and f 00 are not really new coefficients it will be sufficient to say that li li li every term in the perturbation equations is of the form B(y)F (y). This allows us to use another Chebyshev identity [72],
i 1 Xmax 1 B(y)Fl(y) = πl,i Ti(y) − πl,0 (2.77) 2 i=0 2 where i " # Xmax πl,i = bi+j + Θ(j − 1)b|i−j| fl,j (2.78) j=0 with 0 for k < 0 Θ(k) = , (2.79) 1 for k ≥ 0 to expand every term in the perturbation equations in Chebyshev polynomials. Using the definitions Eqs. (2.73)–(2.77) we can also re-write the boundary conditions in Chebyshev form in the same way
2.3.2 Finding κ
With all of the terms in the perturbation equations in Chebyshev form we can extract the Chebyshev coefficients using Eq. (2.71). This will lead to a system of, 2(lmax − 3) + 4imax, algebraic equations for κ and coefficients of the unknown functions, fli . We can schematically represent the system of equations as A(κ)x = 0, where A is a matrix and x is the vector " x = hl hl ... hl ... U l U l ... U l ... W (l+1) W (l+1) ... W (l+1) ... 0 1 imax 0 1 imax 0 1 imax
43 # V (l+1) V (l+1) ... V (l+1) (2.80). 0 1 imax
3 Before finding κ we must incorporate the 2 (lmax − 2) + 3 equations that come from converting the boundary conditions into Chebyshev form. To do this we replace
the equation that came from the highest order extracted coefficient πimax for each eigenfunction with a boundary condition. We solve for κ using the condition det(A(κ)) = 0. This leads to a high degree (O(500+)) polynomial equation for κ. Finding the roots for such a polynomial is difficult when using standard root-finders. Therefore we created two root- finding algorithms which incorporated and went beyond some of the standard root-finding techniques. The key idea of one, is to solve for the roots of the function: tan−1 log | det(A(κ))| instead. The key idea of the other is use the decomposition, SVD(A) = {U, Σ, V}, and find the value of κ that results in the smallest value for last element on the diagonal of Σ. Both root finders achieved convergence at the fourth decimal place for poly- tropic EoS, and the third decimal place for realistic EoS. The loss of precision comes form the fact that realistic EoS have to be numerically interpolated, whereas polytropic EoS have analytical forms, see Sec. 2.5. The first algorithm mentioned was used for the results that appear below, because it was easier to automate. In Fig. 2.1 we show the convergence for an n = 1 polytrope with compactness of .150. Although this figure does show that our code quickly converges, it obscures the
fact that in practice one should increase imax and lmax in step to get convergence.
This figure also shows that the eigenvalue stops converging above lmax = 11 and
imax = 9. For tabulated EoS, a minimum of lmax = 15 and imax = 13 are required
for convergence. Divergence sets in for tabulated EoS at lmax = 21 and imax = 19. The convergence stops after certain values due to truncation errors and the high orders of the polynomials. The issue with both root-finders is that they lead to multiple roots. The way
to determine which root is correct is to start lmax and imax at small values so that one only finds 3 to 5 roots, and then keep the root(s) closest to the Newtonian
estimate of κ0 = 2/3. Then as lmax and imax are increased the correct root will converge whereas the others will change unpredictably.
44 lmax 5 7 9 11 0.5910
0.5908 imax = 9 n = 1 MR = 0.15 0.5906 lmax = 9 Κ
0.5904
0.5902
0.5900 4 5 6 7 8 9 10 11 12
imax
Figure 2.1: Convergence of eigenvalue for n = 1 polytropic EoS. The short (blue) dashes show convergence in imax while fixing lmax = 9, and the longer (purple) dashes show the convergence in lmax when fixing imax = 9. The divergence of the eigenvalue sets for higher values of lmax and imax due to finite precision, see text.
2.3.3 The eigenfunctions
Our procedure for finding the eigenfunctions is as follows. First rewrite all of the variables in the set of equations, A(κ)x = 0, in terms of one variable, e.g. h20 .
Then impose a normalization condition such that h20 = 1 at the surface of the star.
With this solution for h20 at hand, populate the rest of ~x.
There is an issue with solution for the eigenfunctions, hl,Ul,Wl,Vl for stars near the maximum compactness, i.e. the maximum mass of a star stable against radial collapse. Very close to the maximum mass, the solutions develop extra maxima and minima. Evidently the highest-order Chebyshev coefficients are acquiring spuriously high values. We conjecture that this is due to the sensitivity of the stellar equilibrium functions to small perturbations near the maximum mass. This does not seem to significantly affect the eigenvalues though, which continue their smooth trend as functions of compactness.
45 2.4 Results
2.4.1 The n = 1 polytrope
We have successfully reproduced results presented in [49] for an n = 1 polytrope. They found that for a compactness of M/R = .15, κ = .5901. We calculated κ = .5902 for such a star, which represents a difference of only .01%. This small discrepancy could have easily come from the precision of the code that was used in the calculations. Furthermore there is also the difference in design of the code itself. The eigenfunctions we calculated for the κ = .5902 r-mode are exactly the same as those present in Fig. 3 and 4 of [49]. See Figs. below
1
U2 n = 1
MR = 0.15 0.5 10*W3
0
10*V3
10*U4 -0.5 0 0.2 0.4 0.6 0.8 1 rR
Figure 2.2: The U2,V3,W3 eigenfunctions for the κ = .5902 mode
We have extended the results of [49] for an n = 1 polytrope by examining how the r-mode frequency changes as compactness of the star is varied. The results are shown in Fig. 2.4. The compactness was changed by increasing the mass, from
1.01–1.95 M , while holding the radius fixed at 12.53 km. It is clear from the plot that κ decreases as compactness increases. In the plot we include least-squares fits
46 0.05 100*h4
0
-0.05 h2 n = 1
MR = 0.15
-0.1 0 0.5 1 1.5 2 rR
Figure 2.3: The h2, h4 eigenfunctions for the κ = .5902 mode to a linear and quadratic model. The R2 values indicate that the quadratic model is a better fit for the data. It seems that the negative coefficient for the quadratic term is a generic result. The same result was found by [49] for the stellar equilibrium sequence of an n = 0 polytrope. The results for polytropic EoS are a useful guide when examining the results from the realistic or tabulated EoS, because tabulated EoS do not have analytical form like the polytropes but rather one has to use the interpolation methods mentioned in Sec. 2.2.
2.4.2 Tabulated Equations of State
Figure 2.5 contains the values of κ for all 14 EoS under consideration. These were chosen from a standard list of EoS used by [54], and [53], under the constraint that the EoS could support a 1.85 M star, see Sec. 2.1. The lowest mass that was used for any EoS was 1.02 M , the maximum mass used in any calculation was 2.76 M . There are three main things to notice in Fig. 2.5. One, the values of κ decrease as the compactness of the star increases. Two, the generic shape of the
47 κ = 0.616 + 0.352(M R) − 3.47(M R)2 0.60 R-squared: 0.9985
0.58
κ 0.56 κ = 0.719 − 0.864(M R) R-squared: 0.9837 0.54
0.52
0.12 0.14 0.16 0.18 0.20 0.22 0.24
M/R
Figure 2.4: The eigenvalue, κ, for an equilibrium stellar mass sequence of an n=1 polytrope. The radius was kept fixed at 12.53 km, and the mass changed ranged from 1.01–1.95 M . Linear and quadratic fits to the data are also presented. data is parabolic, as shown by the solid (blue) fitted curve. Ordinary-least-squares regression was used to get both the linear and quadratic fit. Examining the R2 value we see again that the quadratic model is a better fit to the data. We also calculated the root mean square error (RMSE) [73], as a way to quantify the deviation of the individual EoS from the quadratic fit. These values are presented in Table I. The total RMSE for all data points is 2.02 × 10−3. Three, the range of κ for tabulated
48 SLy AP3
0.60 AP4 WFF1 WFF2 MPA1 ENG
0.56 MS1 MS1b ALF2 ALF4
κ GNH3
0.52 H4 BBB2 κ = 0.627 + 0.079(M R) − 2.25(M R)2 R-squared: 0.9978 0.48
κ = 0.718 − 0.852(M R) R-squared: 0.9815 0.44
0.11 0.15 0.19 0.23 0.27 0.31
M/R
Figure 2.5: The values of κ for all 14 EoS under consideration. The dashed (red) line represents linear fit, and the solid (blue) line represents the quadratic fit. Ordinary-least-squares regression was used to get both the linear and quadratic fit. The equations for the linear and quadratic fit along with their R2 for the fits are also presented.
EoS is larger than that for n = 1 polytropic model. This is mainly due to the fact that realistic equations are less stiff than the n = 1 polytrope, and so one can squeeze more mass into the same radius, thus increasing the maximum compactness from .220 to .310 for some realistic EoS. This greater range for κ has importance implications for the applications of our results, and these will be explored further in the next section.
49 0.60 AP3 Sly AP4 0.60 AP3 WFF1 WFF2 0.55 0.55 κ κ
0.50 0.50
0.45 0.45
0.11 0.15 0.19 0.23 0.27 0.31 0.11 0.15 0.19 0.23 0.27 0.31
M/R M/R
(a) (b)
Figure 2.6: In (a) we compare the values of κ for EoS derived using variational methods. In (b) we compare the values of κ for SLy and AP3 EoS.
To further examine the results for the tabulated EoS we have split Fig. 2.5 in two plots. Fig. 2.6(a) shows a plot for a family of variational method EoS. The families are grouped together by the techniques used in deriving the equation of state. Figure 2.6(a) shows that the value of κ does not change very much in between members of an EoS family as M/R is varied. Figure 2.6(b) examines whether our results can be used to constrain, or rule out certain EoS. The plots show how the range of κ for SLy [74] relates to AP3 [75]. SLy was chosen because it was one of two EoS that had single member families, and AP3 was choose because it is also present in Fig. 2.6(a). These graphs will be discussed further in the next section. Finally we present two tables. Table 2.1 lists all the tabulated EoS that were used, along with the stable maximum mass of a non-rotating star, the radius of a 1.4M star in the sequence, κ for a compactness of .15, the coefficients for the quadratic fit of the κ values for that EoS, and the RMSE. The R2 value for each fit ranged from .9986 to .9998, again showing that the quadratic model is a good fit for the data. From this fit we see that the quadratic term can be up to a few percent of the frequency. This has significant implications for GW searches, as well as attempts to measure compactness from an r-mode frequency. Table 2.2 gives the numerical values plotted in Fig. 2.5.
50 EoS Mmax R1.4 κ.15 a b c RMSE 1 × 10−3 Sly 2.049 11.736 0.587 0.622 0.151 -2.48 2.35 AP3 2.390 12.094 0.588 0.619 0.142 -2.36 1.39 AP4 2.213 11.428 0.587 0.626 0.150 -2.41 1.56 WFF1 2.133 10.414 0.587 0.617 0.160 -2.40 1.54 WFF2 2.198 11.159 0.587 0.628 0.060 -2.17 1.08 MPA1 2.461 12.473 0.588 0.629 0.052 -2.14 1.70 ENG 2.240 12.059 0.588 0.613 0.215 -2.56 1.51 MS1 2.767 14.918 0.588 0.624 0.107 -2.30 1.62 MS1b 2.776 14.583 0.589 0.619 0.156 -2.40 1.98 ALF2 2.086 13.188 0.588 0.632 0.026 -2.08 1.63 ALF4 1.943 11.667 0.587 0.632 0.002 -1.98 1.25 GNH3 1.962 14.203 0.587 0.639 -0.027 -2.09 2.80 H4 2.032 13.774 0.591 0.640 - 0.002 -2.13 3.65 BBB2 1.918 11.139 0.587, 0.611 0.249 -2.71 2.83
Table 2.1: We present a list of all the tabulated EoS, for each EoS we show the stable maximum mass, the radius for a 1.4M star, κ for a compactness of .15, 2 M M the coefficients for the quadratic fit of the κ of the form a + b R + c R , and the root mean square error (RMSE) of the EoS data points to the quadratic fit.
2.5 Discussion
We begin with a discussion of the n = 1 polytrope results. The fact that the r-mode frequencies go down as compactness increases might come as a surprise. However, one must remember to keep track of reference frames. In the reference frame of the star κ does decrease because the restoring force is proportional to ω¯(r) = Ω − ω, and this goes down as compactness is increased. But the observed frequency in the inertial frame is |κ − m|Ω, see Eq. (2.3), and this increases as κ decreases. To compare the results from the n = 1 polytrope, and tabulated EoS we can examine Fig. 2.4 and 2.5. From this we notice that the κ for tabulated EoS can differ from those of the polytropic model by of order ten percent in frequency. This shows the need to use realistic EoS when calculating r-mode frequencies. Let us now discuss the implications of the parabolic shape of the κ values shown in Fig. 2.4, and 2.5. Overlaid on these plots are two fits from which it is clear that the quadratic fit is better than the linear one. These figures show that the
51 corrections to the r-mode frequency using post-Newtonian approximations must be carried out to at least second order. For example, the first order post-Newtonian formula for an n = 0 polytrope given by [49] is
2 " 8(m − 1)(2m + 11) M # κ = 1 − , (2.81) pN m + 1 5(2m + 1)(2m + 5) R for m = 2 this gives 2 8 M κ = 1 − . (2.82) pN 3 15 R Figure 2.4 and 2.5 show that the equation above is insufficient since it does not account for quadratic, (M/R)2, corrections. Moving on to the astrophysical applications of our results. Gathering data from realistic EoS we see that the range of κ is approximately 0.614–0.433 for compactness values 0.110–0.310. If the spin frequency of a star is known from electromagnetic observations, this range can be used to conduct narrow-band gravitational wave searches for known pulsars. Using Eq. (2.3) we see that our range of κ gives the range 1.39Ω < σI < 1.57Ω, where σI is the frequency observed in our reference frame. Taking the Crab pulsar (Ω/2π = 29.7 Hz, [58]) as an example, our range of σI suggests a narrow-band search for r-modes be carried out from 41.3–46.6 Hz. Alternatively, our results will be of use in the case where a GW detection is made for an electromagnetically unknown pulsar. Suppose the gravitational wave signal has a frequency of 100 Hz. Astronomers can search for the pulses at 50 Hz, assuming the signal came from a non-axisymmetric deformation in the star, and in the range of 63.7–71.9 Hz, assuming the signal came from r-modes of the star. Another potential use for this research that was mentioned in the introduction was the ability use the r-mode frequencies to constrain the nuclear equation of state. We see from Fig. 2.6(a) that r-mode detections alone will not be enough to distinguish between members of an EoS family. Figure 2.6(b) shows a slightly more promising result. This figure shows that it may be possible to distinguish between different EoS families. However, this would be made difficult without additional electromagnetic data on quantities such as the compactness. This is due to the scale of the deviations seen in Fig. 2.6(b), which are less than 1%. These deviations to κ have to compete with the physical phenomena described in Sec. 2.1 which for
52 most stars can also change κ by 1%. Therefore it will be impossible to distinguish what is truly giving rise to the change in κ without more research into these effects. In this context it is interesting to note the recent report in [43] of the possible detection of an r-mode in the outburst of the accreting millisecond pulsar XTE J1751-305. Reference [45] showed that the observed frequency of the oscillation, if interpreted as the r-mode frequency, gave rise to a sensible constraint on the mass-radius relation for the star, see their Fig. 1. Reference [45] included both relativistic corrections and rotational ones in their analysis, but the former were based on the uniform-density calculations of [49], rather than realistic EoS of the sort considered in this paper. However, comparison of the plot of κ verses compactness in Fig. 1 of [45] with Fig. 2.5 above shows that the analysis would not change significantly if it were repeated using realistic EoS, something to be expected given the rather narrow variation of κ with EoS shown in our Fig. 3. The situation is broadly similar for the second possible neutron-star r-mode detection [44]. To sum up, we have been successful in finding a range of r-mode frequencies for both polytropic and tabulated EoS. Furthermore we have shown that our results can be used as input data for electromagnetic and GW searches. Along with these successes there are some issues that should be discussed. One issue we encountered is that the precision of the κ values decreased by an order of magnitude for the tabulated EoS compared to polytropes. We believe this comes from the fact that the tabulated EoS have to be interpolated instead of coming in analytical form like polytropes. From our discussion in Sec. 2.2.1 we know that the different interpolation schemes results in percent differences 0.1%. This shows that when using tabulated values we can only hope to get up to three significant digits, regardless of the interpolation scheme used. It is difficult to address this issue since it is an inherent problem with tabulated EoS. Perhaps in future work we can use analytical fits to the tabulated EOS such as those presented in [53] and [54] to mitigate this issue. Another issue is of course the various physical phenomena that were ignored. These will inevitably have some impact upon the r-mode frequency, even if our simple estimates indicated that relativistic effects were likely to be the most important. Therefore it is still necessary to explore the effects of other mechanisms. In particular, in the future we would like verify that the corrections that come from rapid rotation are significant only for very rapidly rotating stars, and that the
53 crustal effects are important only in narrow spin frequency bands. Including these effects would allow us to distinguish between families of EoS if we get compactness information.
2.6 Conclusion
In this chapter we calculated the r-mode frequencies for slowly rotating neutron stars which were modeled using realistic equations of state. Before discussing the search for r-modes from super nova remants using data from the LIGO detectors (see Ch. 4 ), we first describe the general set of data analysis tools used in gravitational wave detection.
54 M/R SLY AP3 AP4 WFF1 WFF2 MPA1 ENG MS1 MS1B ALF2 ALF4 GNH3 H4 BBB2 0.11 ------0.612 0.616 - 0.12 0.606 - - - - 0.606 - - - 0.606 - 0.606 0.610 - 0.13 0.600 0.600 0.600 0.599 0.600 0.600 0.600 0.601 0.601 0.600 0.599 0.600 0.604 0.599 0.14 0.594 0.594 0.594 0.593 0.594 0.594 0.594 0.595 0.595 0.595 0.593 0.595 0.597 0.594 0.15 0.588 0.588 0.587 0.587 0.587 0.588 0.588 0.588 0.589 0.588 0.587 0.587 0.591 0.587 0.16 0.581 0.581 0.581 0.581 0.581 0.583 0.582 0.582 0.58 0.581 0.581 0.581 0.586 0.581 0.17 0.574 0.575 0.574 0.574 0.574 0.575 0.575 0.576 0.576 0.576 0.574 0.574 0.58 0.574 0.18 0.567 0.568 0.567 0.567 0.567 0.567 0.568 0.57 0.569 0.569 0.568 0.566 0.573 0.567 0.19 0.56 0.561 0.56 0.56 0.56 0.561 0.561 0.558 0.562 0.564 0.56 0.56 0.564 0.559 0.20 0.553 0.553 0.551 0.552 0.553 0.554 0.553 0.554 0.553 0.556 0.553 0.551 0.556 0.552 0.21 0.544 0.545 0.544 0.544 0.545 0.545 0.545 0.546 0.546 0.546 0.545 0.542 0.545 0.544
55 0.22 0.535 0.537 0.536 0.536 0.536 0.538 0.537 0.537 0.538 0.538 0.536 0.532 0.536 0.535 0.23 0.526 0.528 0.527 0.527 0.527 0.529 0.528 0.53 0.531 0.529 0.527 0.523 0.528 0.526 0.24 0.516 0.519 0.518 0.518 0.518 0.519 0.518 0.518 0.519 0.519 0.518 0.511 0.518 0.516 0.25 0.506 0.509 0.508 0.508 0.508 0.51 0.509 0.509 0.51 0.508 0.509 - 0.512 0.505 0.26 0.495 0.498 0.497 0.498 0.497 0.498 0.498 0.498 0.499 0.498 0.497 - - 0.494 0.27 0.482 0.488 0.486 0.487 0.485 0.488 0.486 0.486 0.488 - - - - 0.482 0.28 0.469 0.475 0.472 0.475 0.473 0.475 0.474 0.472 0.474 - - - - 0.468 0.29 0.454 0.462 0.46 0.463 - 0.463 0.461 - 0.461 - - - - 0.453 0.30- - 0.448 0.445 0.449 - 0.447 0.444 ------0.31- - - - 0.433 ------
Table 2.2: This table gives the numerical κ for all tabulated EoS, over the range of compactness values considered in our analysis. These values are plotted in Fig. 2.5. The “-" indicates a compactness that could not be obtained with that EoS. Chapter 3 | Fundamentals of gravitational wave data analysis
3.1 Introduction
In Ch. 1 we gave a brief summary of gravitational wave detection using LIGO. We described the idea behind the experiment, and in Fig. 1.8 showed the sensitivity of the interferometers used in the experiment. Of course the sensitivity curves do not tell the whole story. In general, the amplitude of a continuous wave signal will be weaker than the noise of the detectors. Therefore special data analysis techniques such as matched filtering ([30,31], Sec. 1.4.3) are necessary. The technique relies on the fact that continuous wave signals can be a priori modeled to great accuracy. The basic idea behind matched filtering is to coherently integrate this model waveform with the data and use this to test the hypothesis of whether a signal is present in the data. There are certain issues that go along with using the matched filter approach, such as how to get the waveform model, this is discussed in Sec. 3.2. Another issue is what decision statistic should be used with the coherent integration to decide if the data was a real signal or just noise, this is discussed in Sec. 3.3. A real GW signal will depend on many continuous parameters, but this would lead to an infinite amount of filters (templates) that must be checked. In a real search we will only have a limited number of templates to use and thus we need to find out how to efficiently place these templates, so that the minimum amount of templates are used to cover the volume of the parameter space. The algorithm to generate such a
56 “template bank” is discussed in Sec. 3.4. Finally there is the added computational cost of using this method because the integration is done coherently. A resampling method that can be used to reduce this computational cost is discussed in section Sec. 3.5.
3.2 Signal model
We first address the issue of how to model the signal that is to be used for the coherent integration with the data. Further details on material in this section can be found in [30]. The signal is modeled as a plane-wave solution as presented in Eq. 1.10. The
main task is to convert this solution from the reference frame of the wave (xw, yw, zw)
into the detector frame (xd, yd, zd). We can write Eq. 1.10 slightly differently as
+ × Hw(t) = h+Hw + h×Hw , (3.1)
where 1 0 0 + Hw = 0 −1 0 (3.2) 0 0 0 and 0 1 0 × Hw = 1 0 0 . (3.3) 0 0 0
The orientation of the xw-yw plan with respect to the zw-axis is given by the polarization angle ψ. The individual components are
h+(τ) = A+ cos Φ(τ) h× = Ax sin Φ(τ) (3.4)
where A+ and A× represent amplitudes and Φ(τ) is the phase of the GW in the proper time of the source, given by τ. The GW in the detector frame is
T Hfd(t) = M(t)Hw(t)M(t) , (3.5)
57 where M is the three-dimensional orthogonal matrix which transforms the coordi- nates from the GW frame to the detector frame. This matrix is broken up into T M = M3M2M1 , where M1 is a transformation from the wave coordinates to the
celestial sphere, M2 is a transformation from celestial to cardinal coordinates, and
M3 is a transformation from cardinal to detector’s coordinates. In the detector’s frame the gravitational wave signal is
1 h i 1 h i h(t) = n · Hf (t)n − n · Hf (t)n , (3.6) 2 1 d 1 2 2 d 2
where n1 and n2 are,
n1 = (1, 0, 0) , n2 = (cos ζ, sin ζ, 0) . (3.7)
They represent the unit vectors parallel to the arm number 1 and 2 of the detector.
The order of arms is such that the vector n1 × n2 points outwards from the surface of the Earth [30], and ζ is the angle between the two arms. For the LIGO and Virgo detectors ζ = 90◦. The signal in Eq. 3.6 is usually re-written in terms of the detector’s antenna
(beam-pattern) functions F+,F× (see [30]), as
h(t) = F+(t)h+(t) + F×(t)h×(t), (3.8)
where F+ and F×
F+(t) = sin ζ [a(t) cos 2ψ + b(t) sin 2ψ] , (3.9)
F×(t) = sin ζ [b(t) cos 2ψ − a(t) sin 2ψ] , (3.10)
where a(t) and b(t) are
1 a(t) = sin 2γ(3 − cos 2λ)(3 − cos 2δ) cos[2(α − φ − Ω t)] 16 r r 1 − cos 2γ sin λ(3 − cos 2δ) sin[2(α − φ − Ω t)] 4 r r 1 + sin 2γ sin 2λ sin 2δ cos[α − φ − Ω t] 4 r r 1 − cos 2γ cos λ sin 2δ sin[α − φ − Ω t] 2 r r
58 3 + sin 2γ cos2 λ cos2 δ, (3.11) 4 b(t) = cos 2γ sin λ sin δ cos[2(α − φr − Ωrt)] 1 + sin 2γ(3 − cos 2λ) sin δ sin[2(α − φ − Ω t)] 4 r r + cos 2γ cos λ cos δ cos[α − φr − Ωrt] 1 + sin 2γ sin 2λ cos δ sin[α − φ − Ω t], (3.12) 2 r r where λ is the latitude of the detector, Ωr is the rotational angular velocity of the
Earth, φr is defined as the phase which determines the position of the Earth in its diurnal motion at t = 0 (the sum φr + Ωrt, coincides with the local sidereal time), γ determines the orientation of the detectors with respect to the local geographic coordinates, and is measured from East to the bisector of the detector arms, and α and δ are the right ascension and declination of the GW source [30]. The phase of the signal can be changed from the proper time of the source, τ, to the detector time t. The full derivation is show in [30] with a summary given in [76], and the result is:
s k+1 s k X (k) t 2π X (k)t Ψ(t) = Φ0 + 2π f + n0 · rd(t) f , (3.13) k=0 (k + 1)! c k=0 k!
(k) where f is the kth time derivative of the instantaneous frequency evaluated at t = 0 at the solar system barycenter (SSB), n0 is the constant unit vector in the direction of the Sun and rd is the position vector of the detector in that frame.
It should be noted that A+ and A× are not constant amplitudes, but in fact depend on the the initial phase of the signal φ0, and the polarization angle ψ. The signal in Eq. 3.6 can be rewritten as
4 X h(t, A, λ) = Ai h(t, λ) (3.14) i=1 where the amplitudes Ai are
A1 = A+ cos φ0 cos 2ψ − A× sin φ0 sin 2ψ (3.15)
A2 = A+ cos φ0 cos 2ψ + A× sin φ0 cos 2ψ (3.16)
A3 = −A+ sin φ0 cos 2ψ − A× cos φ0 sin 2ψ (3.17)
59 A4 = −A+ sin φ0 sin 2ψ − A× cos φ0 cos 2ψ (3.18) where λ contains the the Doppler parameters: the sky position, the frequency, and spin-downs. The model shown in Eq. 3.14 is the one that will be used in the coherent integration and calculation of the F-statistic.
3.3 Maximum likelihood detection
The matched filtering method relies on a statistical hypothesis test. Given the data stream x(t), our null hypothesis is that the data only contains noise n(t). The alternative hypothesis is that the data contains noise and a GW signal h(t, A, λ). The statistic used to determine the validity of the null hypothesis is called the F-statistic. In order to derive this test statistic we start with the log of the likelihood ratio Λ, which represents the relative probability of x(t) containing signal in addition to noise to the probability that it only contains noise:
p(x|h + n) ln Λ = (3.19) p(x|n) where 1 p(x|n) = N exp[− (x||x)] (3.20) 2 and 1 p(x|n + h) = N exp[− (x − h||x − h)]. (3.21) 2 We define Z ∞ −1 ∗ (x||y) = 4R dfx˜(f)Sn (f)˜y(f) (3.22) −∞ where Sn(f) is the one sided power spectral density, x˜ represents the Fourier transform of x, and ∗ represents complex conjugation. The likelihood can be simplified to 1 ln Λ = (x||h) − (h||h) . (3.23) 2 We want the probability of p(x|n + h) to be larger than p(x|n), i.e. we would like ln Λ to be at its maximum value in order to claim a detection. To find the values
60 of the signal model A, λ which gives the maximum value for the likelihood ratio we need to the find the solutions of
∂ ln Λ = 0 (3.24) ∂Ai and ∂ ln Λ = 0 . (3.25) ∂λ
The likelihood depends linearly on Ai and so it is straightforward to maximize over these by evaluating 3.24. We get four linear equations for Ai, with the solutions
4 max X −1 Ai = [M ]ij(x||hj), (3.26) i=1 where
Mij = (hi||hj) (3.27) and M −1 is its inverse. These inner products are computed over many cycles of the signal and so the values average out as M11 ≈ M33, M22 ≈ M44, M12 = M21 ≈
M34 = M43, all others are zero. Plugging the solutions of these equations back into Eq. 3.23 gives
4 4 X max X −1 2F = Ai (x||hi) = [M ]ij(x||hi)(x||hj) . (3.28) i=1 i=1
If we expand these terms we get exactly the F-statistic derived in [30],
B(x||h )2 + A(x||h )2 − 2C(x||h )(x||h ) 2F = 1 2 1 2 D B(x||h )2 + A(x||h )2 − 2C(x||h )(x||h ) + 3 4 3 4 , (3.29) D where
A ≈ 2(h1||h1) ≈ 2(h3||h4),B ≈ 2(h2||h2) ≈ 2(h4||h4) 2 C ≈ 2(h1||h2) ≈ 2(h3||h4),D = AB − C . (3.30)
It is not possible to expand h(t) linearly in λ and thus trying to analytically maximize over these variables is a difficult task. In order to find a signal what we
61 must do is create a parameter space that spans the Doppler variables (λ) and search over them. Each point in the parameter space represents a GW waveform for which we calculate a value of 2F. If the waveform gives a statistically significant value then it will be chosen for further analysis. The methods by which we cover the parameter space with these model waveforms (or templates) are discussed in the next section. For now we finish by discussing how to establish what a statistically significant event means.
If there is no signal present the four inner products (x||hi) are correlated Gaussian random variables. Reference [30] shows a transformation by which the correlation can be undone and in this case the F-statistic can be written as
1 F = (z2 + z2 + z2 + z3) , (3.31) 2 1 2 3 4
1 where zi is a Gaussian random variable . This means that the values for 2F will be drawn from a χ2 distribution with four degrees of freedom. If a signal is present the 2F are still drawn from a χ2 distribution but it is centered around
d2 = (h||h) , (3.32) where d2 is referred to as the optimal signal-to-noise ratio. A template is marked for further investigation if its 2F value is beyond a threshold. The threshold is usually set by the condition that the 2F value would have a less than 5% chance of being picked randomly. Finally, we note that the F-statistic can be generalized to multiple detectors as shown by [77], by replacing the time series x(t) with a vector value time series ~x(t), and re-defining the inner product (x||y) as
Z ∞ ˜ −1 ˜ ? (~x||~y) = 4R df~x(f)Sn (f)~y(f) (3.33) −∞
−1 where Sn (f) includes the cross-correlation of the noise in pairs of detectors. 1Notice the factor of one-half, this is why it is standard to quote 2F values
62 3.4 Template spacing
A fundamental question that arises when using the matched filtering technique is how many matches need to be used. That is, how many model signals h(t) do we need to integrate with the data x(t). The signal depends upon physical parameters such as the wave amplitude, the wave polarization, initial phase, and the inclination angle; these are the so-called nuisance parameters. We do not search over the nuisance parameters because the likelihood ratio is already maximized with respect to these parameters, see 3.24. The signal also depends on the right ascension and (k) declination, the frequency, and spin-down parameters f of the source, where the k superscript represents the k-th time derivative. For the directed searches described in the following chapter, the parameter space which is searched depends on f, f,˙ f¨, because the sky-location is known. The physical parameter space has of course an infinite number of points as each of these parameters is a continuous variable. However in terms of creating filters, we seek the smallest parameter space possible so as to decrease computational cost. Therefore, we must find an efficient way of picking these points so that we can make GW detections but avoid the searches being computationally prohibitive.
3.4.1 Sphere-covering
The solution is to realize that this problem of efficiently picking points in the parameter space can be mapped to a sphere covering problem found in condensed matter physics. The solution is presented in much greater detail in [76]. A sphere covering is a set of n-dimensional spheres, each centered on a member of a set of points P in a n-dimensional space Rn. The points will form a lattice L which can be generated by a matrix G. The lattice has a property called thickness denoted by θ, which defines the average number of spheres enclosing any point in Rn. The closer θ is to unity the more efficient the covering, in the sense that the least amount of spheres possible is used to covered the largest possible volume. It should be clear why the sphere packing problem can be mapped onto the problem of generating the parameter space (template bank) for our directed searches.
63 3.4.2 Template Spacing Metric
To use the sphere covering problem to solve the problem of template bank generation we have to transform the distance between templates to distances between lattice points. The distance between templates is determined by the maximum amount of loss in signal to noise ratio (SNR) allowed by our search. A 20% loss in SNR is acceptable for continuous wave searches. This is done to balance the cost of conducting a search with the computational resources available, since the cost scales steeply as the loss in SNR is reduced. Therefore the cost is initially kept small and if there was a suspected signal in the data we preform a follow up search with a finer template spacing. Using this condition we can determine the distance between templates. In order to measure the loss in SNR we will use the detection statistic discussed in Sec. 3.3. This statistic is proportional to the SNR if we assume one template is the signal and another template is to be used as a match. The expectation value of 2F represents the likelihood of detecting a signal, h(A, λ, t) in the data x(t). If the data contains a perfect match h(λ), then 2F(λ) will be a maximum value. At a nearby point, λ0 = λ + ∆λ, 2F(λ) will have a lower value. If we assume ∆λ is small we can expand 2F(λ) as
1 2F(λ) = 2F(λ0) + ∆λH∆λ (3.34) 2 where ∂2(2F) [H]ij = , (3.35) ∂λi∂λj and the first derivative vanishes because 2F(λ) is a maximum. Now let us define the mismatch from Eq. 3.34 as
2F(λ) − 2F(λ0) µ(λ, λ0) = 2F(λ) i j = Mi,j∆λ ∆λ (3.36) to be the loss of SNR, i.e. the decrease in 2F due to mismatch between Doppler parameters λ and λ0. Here H M = − i,j . (3.37) i,j 2F(λ) We have intentionally written Eq. 3.36 to suggest that µ(λ, λ0) represents the
64 distance between two templates centered around λ, and λ0 respectively, and that
Mi,j represents the metric on this space of templates. To construct a template bank we require the mismatch to be bounded by a maximum value, µmax. Pictorially, the parameter space is filled by ellipsoids centered around λ with axis given by
s µmax ai =v ˆi (3.38) vi where vˆi and vi are the eigenvectors and corresponding eigenvalues of Mi,j.
The full metric Mi,j actually depends upon the Doppler parameters and A, and scales with Tspan, i.e. the amount of data that is integrated. If the value of
Tspan is more than several days then the metric is simplified to a phase metric:
* + * + * + Φ ∂Φ(t) ∂Φ(t) ∂Φ(t) ∂Φ(t) [M ]i,j = − . (3.39) λi λj λi λj
This metric has been computed for the single-detector F-statistic by [78]. It was shown by [77] this metric does not scale with multiple detectors, i.e. this metric can be used for both the single-detector F-statistic and multi-detector F-statistic. The phase metric for parameters α, δ, f, f,˙ f¨, is not in general flat. However it was shown by [79] that if the sky-position is known, and the metric only depends on f, f,˙ f¨, etc. then it is flat. Therefore in direct searches the metric is given by
4π2T i+j+2(i + 1)(j + 1) [M f,f,...˙ ] = span , (3.40) i,j (i + 2)!(j + 2)!(i + j + 3) where the indices i, j take the values 0 for f, 1 for f˙ and so on. This is a good time to note that the overall computational cost for running a gravitational wave directed search scales with Tspan and the various frequency domain parameters. The scaling is
2 2 2 3 n+1 Tspan fmaxTspan fmaxTspan fmaxTspan fmaxTspan . . . fmaxTspan ... (3.41) | {z } | {z } | {z } | {z } | {z } α, δ f f˙ f¨ f (n)
Looking back at the phase metric Eq. 3.40 we can see why the scaling works this ¨ 4 way. If we search over f then the Tspan in that equations goes as T , since i = 0
65 3 and j = 2. From the cost scaling equation above we would get Tspan × fmaxTspan. Thus for a directed search where we know the sky-position, but have to search over ˙ ¨ 3 7 f, f, f, the overall scaling is fmaxTspan.
3.4.3 Lattice generation
The template space points ~p, are calculated using the lattice generator G and an n-dimensional vector of integers ~k, i.e. ~p = G~k. For our directed searches the template space points are given by f, f,˙ f¨, and these correspond to the points p. The range of values for f˙ is bounded by f, and the range of f¨ is bounded by f˙. This physical restriction on the points ~p must be converted into a equivalent restriction on the lattice points ~k. In order to ensure this restriction is met G must be a lower triangular matrix. To perform the transformation of the lattice points to the parameter space points we start with points q and q0 on the surface of a covering sphere of radius R, then (q − q0) · (q − q0) = R2. (3.42)
Then we define a transformation, ~p = T~q √ µ T = max D , (3.43) R where the matrix D has columns which are orthonormal to the matrix Mi,j, i.e. DT MD = 1. Then the square of the metric distance between p and p0 is given by
µ h i (p − p0) · M(p − p0) = max (q − q0) · DT MD(q − q0) = µ (3.44) R2 max
Which means the p~0 is on the surface of the metric mismatch ellipse sourrounding
~p. Given any m × n generating matrix GL, we have √ µ G = max DKJQT G L , (3.45) R L
T where Q GL is used for the QR decomposition, J is the identity matrix with zero padded columns, K and L will reduce the matrix in between them into a lower triangular form. And finally the numerical factor times D is the transformation. Having completed the summary of how to generate the template bank we move
66 onto a discussion of the resampling algorithm and how it affects the computational cost of a GW search.
3.5 Resampling
Calculating the F-statistic is computationally inexpensive, but it does get costly for a GW search because one has to coherently integrate the detector’s output data, x(t), with approximately 1013 templates. This greatly limits the amount of data that can be analyzed. In order to speed up the calculations researchers have made clever use of numerical techniques and innovations in technology [80]. In calculating the F- statistic one must compute the integrals in Eq. 3.29. In the previous versions of the pipeline this has been done using the Dirichlet kernel, which is
n n X ikx X sin((n + 1/2)x)) Dn = e = 1 + 2 cos(kx) = . (3.46) k=−n k=1 sin(x/2)
The convolution of Dn with any 2π-periodic function represents the n-th degree Fourier approximation to the the function. In the past, the convolution was carried out to 16 terms. However when trying to utilize Intel’s SSE2 floating point extension to speed up various calculations it was discovered that the 16 term Dirichlet kernel proved problematic and so the convolution is now carried out to 8 terms. The Dirichlet kernel being an order N 2 operation, this reduction in terms leads to an overall speed up of a factor of 4. In this section we discuss yet another way to speed up the calculation of F-statistic. The idea for resampling was first presented in [30] and its first imple- mentation is discussed in [81]. In this method the usage of the Dirichlet kernel is stopped and the integrals are instead calculated using Fast Fourier Transforms (FFTs). The SSE2 extensions are still utilized. The key insight of resampling is that the change in time coordinate of the Earth-based LVC detectors is almost equal (up to small perturbative corrections) to the change in the time coordinate of the solar system barycenter (SSB) frame
see Fig. 3.1 To see why this is the case, we define a new time, tb, as the SSB time coordinate which is
tb = t + tm . (3.47)
67 Figure 3.1: This figure shows the time differences used in resampling. SSB is the solar system barycenter. The time elapsed on the detector is t and that of the SSB is tb. This figure is from [81]. where n · r (t) t = 0 d (3.48) m c and n0 is the unit vector pointing to the source in the SSB frame, and rd is the position vector of the detectors in the SSB frame. Now if we take the time derivative of Eq. 3.47 we get dt dt b = 1 + m (3.49) dt dt and dt n · v (t) m = 0 d , (3.50) dt c
68 where vd is the velocity of the detectors in the SSB frame, so it represents the Doppler shift of the source with respect to the detector. For Earth-based detectors −4 this is approximately 10 , and so we have from Eq. 3.49 that dtb ≈ dt. This approximation is the crux of the resampling technique. It means that instead of accounting for the Doppler modulation of the signal due to the Earth’s rotation, we can change to a new time coordinate and forget about these specific Doppler corrections altogether because they are insignificant. This motivates the definition for the phase of GW source as
(k) Φ(t) = 2πf[t + tm(t; α, δ)] + Φs(t; f , α, δ), (3.51)
where Φs = Ψ(t) − Φ0, see Eq. 3.13. This definition explicitly has the detectors modulation due to the rotation about the SSB separated from the modulation due to the intrinsic frequency. If we also define
Z T0 2 −iΦ(t) Fa(f) = a(t)x(t)e dt (3.52) −T0 2 and Z T0 2 −iΦ(t) Fb(f) = b(t)x(t)e dt , (3.53) −T0 2 then the F-statistic can be expressed as
4 B|F |2 + A|F |2 − 2CR(F F ∗) F = a b a b . (3.54) Sh(f)T0 D
By using the resampling argument above we can rewrite the integrals for Fa and
Fb as T0 Z 2 −2πiftb iΦs(tb) Fa(f) = a(tb)x(tb)e e dtb (3.55) −T0 2 and T0 Z 2 −2πiftb iΦs(tb) Fb(f) = b(tb)x(tb)e e dtb , (3.56) −T0 2 which are Fourier transforms of the data x(tb), and the detector response a(tb), b(tb) functions resampled in the SSB frame multiplied by a phase eiΦs(tb). Computing Fourier transforms is a much quicker way to evaluate these integrals versus the
69 Dirichlet kernel method. The FFT is a N log N computation whereas the Dirichlet kernel is N 2.
3.5.1 Implementation
It would seem that the resampling method described would be implemented on time-series datasets. However, the calculation of 2F actually proceeds in the frequency domain. The reason being, much of the data analysis code used by the LVC is written with frequency domain analysis in mind because working in the frequency domain is one way to manage the noise sources in the detectors. Due to the discrepancy in how the algorithm is implemented versus how it should be naturally implemented, there are extra steps of overhead that must be done. This overhead comes from taking data already in the frequency domain and changing it to time-series data, and then changing it back. Let us give a brief overview of the resampling algorithm. The first step is to combine the SFTs given a Tspan and frequency range into one long-baseline SFT. The next step should be to invert FFT this SFT and get a time-series data, then resample the time-series in the SSB frame and calculate Fa and Fb. However, there are extra steps taken before the inverse FFT to reduce the computational cost of the inverse FFT. A simple way to reduce the cost of the FFT is to reduce the number of points used. This is known as downsampling. The down-sampled data comes from picking every Dth point of the original dataset. In order to know which points to keep a procedure called heterodyning is used. Heterodyning simply amounts to shifting the frequency band of the data. For example if we suspect a signal from a dataset from zero to 1kHz lives just in the band from 990 Hz to 1kHz we can shift 990 Hz to -5 Hz and 1kHz to 5 Hz. This is an example of heterodyning. This process leads to no loss of information as we really just relabeled the frequency bins. There are however large implications of this shift. This shift will change the Nyquist frequency of the data. This frequency comes from the Nyquist-Shannon sampling theorem [82]. This theorem states that if we wish to resolve a signal at frequency n Hz, then the rate at which our instrument takes measurements must be 2n Hz. Thus in the example above where we had data sampled from zero to 1kHz, the Nyquist frequency is 2kHz. If we shift the frequency to 5 Hz the new
70 Nyquist frequency is 10 Hz. The thing that stayed the same was the bandwidth (B) of the data, this is 10 Hz both times. The bandwidth, the new Nyquist frequency, the old Nyquist frequency, and D can all be related by the following equation
f B = 2f = 2 Nyq,old (3.57) Nyq,new D
Using this in the example above with B = 10 Hz, and fNyq,old = 1 kHz we can solve for D which is 100. This means that to down-sample that dataset we would select one out of every hundred points. One caveat of downsampling is aliasing. Aliasing refers to the ambiguity that
arises when trying to identify a signal at frequency f0 from its harmonics f ± kf0, where k ∈ Z, when dealing with discrete data. In order to avoid this issue the heterodyned data are first low-pass filtered. The steps taken by the algorithm are as follows. Take a dataset in the frequency domain, and create long-baseline SFT. Then heterodyne the data, apply a low-pass filter and down-sample the data. Then inverse FFT the long-baseline SFT
and produce a time-series dataset. Produce the time-series functions, x(tb), a(tb),
and b(tb) by interpolation using cubic splines. Finally, transform back into the frequency domain and compute the F-statistic.
3.5.2 Speed up
The main aim of the resampling algorithm is to speed up the calculation of the F- statistic. This statistic is calculated for every template in our parameter space, and (k) it requires looping over the sky-location2 and spin-down parameters f . Both the old implementation of the software to compute the F-statistic, compute_FStatistic_v2 (CFS) and the new resampling implementation which we refer to as CFS_resamp loop over these parameters. The main difference is how the two algorithms loop over frequency. If we assume N data points (templates) and that the number of operations
per sky-location and spin-down is Nops, then the number of operations done by the
2For our directed searches, the spin-down parameters are looped over, but this does not change the argument presented in this section.
71 Dirichlet kernel method of calculating F-statistic as used in CFS is given by
CFS NT ot = Nops · NDir · NSF T s · N (3.58)
where Tobs NSF T s = . (3.59) TSFT
In CFS_resamp there are 4 main steps, calculate tb(t), calculate Fa and Fb, inter- polate beam patterns, and take the Fourier transform. Each of the first three steps has approximately 10 steps, and since they are done sequentially this adds up to 30 operations. The fourth step takes N log N operations. Therefore the total number of operations for CFS_resamp is
Resamp Resamp NTot = (Nops + log N) · N. (3.60)
Therefore the ratio of operations between the two methods is
LAL NTot Nops · NDir_Ker · NSFTs Resamp = Resamp (3.61) NTot Nops + log N
To first order, we have LAL NTot NSFTs Resamp ≈ . (3.62) NTot log N
3.5.3 Issues
The first issue with resampling is that the speed estimate is not completely accurate. Indeed to first order in so-called big-O notation Eq. 3.62 is true. However in reality
it is important to keep the linear term. For example, the Tspan used for the two searches discussed in the next chapter are 24 days (≈ 2 × 106 s) and 51 days (≈ 4 × 106 s). If we assume the data was sampled at 100 Hz, then we have ≈ 108
data points. Looking at the linear term in Eq. 3.60, which scales as log2 N, we 8 have log2 10 ≈ 27 computations. If we consider instead the number of templates 12 12 for a search ≈ 10 , then log2 10 data points ≈ 40 computations. In both cases the linear term is of the same order as the 30 resampling operations. Therefore when trying to use the CFS_resamp to estimate the cost for a computation one must keep the linear term. This issue resulted in problems for our GW searches
72 discussed in the next chapter. Another issue with the resampling algorithm is the interpolation phase. The interpolation is done because the time-series points need to be down sampled. The downsampling is done so that the overhead computational cost of Fourier transforming a large time-series dataset is shortened. The key problem with the spline interpolation scheme used is one that plagues many interpolation schemes, that is, errors at the boundaries. In the current implementation of resampling, the solution is to simply double the frequency band one is searching over. This way the errors occur at boundaries that are physically unimportant. This simple solution works because the speed up gained in doing CFS_resamp over CFS can be large. However, this doubling of the frequency band of interest did cause problems for the two searches discussed in the next chapter. The issues arose when software injected GW signals were being used to place upper limits on GW emission from our sources. The injected signals have narrow frequency bands, typically 10−5 Hz. Resampling doubles this and because there are twice as many bands to check over the overhead cost of the FFT, its inverse, heterodyning, low-pass filtering, and downsampling, are all doubled. The problem isn’t as simple as this because the doubling of the frequency band does not just mean the upper-limit injections take twice as long to do, in fact they take approximately 80 times longer.
73 Chapter 4 | Using barycentric resampling in gravitational wave searches from two supernova remnants
4.1 Introduction
The LVC has conducted several searches for gravitational waves (GWs). The first search for GWs was conducted on a single pulsar, whose frequency, frequency derivatives (spin-downs), and sky-location (right ascension and declination) were all known [83]. These are known as targeted searches. Because so much is known about the sources these searches are not computationally expensive and allow for the best sensitivity for a given span of data. Thus far 195 pulsars have been used as sources for a targeted search [84–88]. Another class of searches is the all-sky search. The idea here is to cover a wide range of sky-locations and spin-down parameters and search for GWs from neutron stars not seen as pulsars [89–96]. These searches are not as sensitive as the targeted searches and due to the large parameter space that must be covered are very computationally intensive (expensive). The searches described in this work balance the sensitivity of the targeted search and the computational cost of the all-sky search. We aim to search over frequency and spin-down parameters for sources whose sky-position is known. These are called directed searches. The first directed search was conducted for an accreting neutron star in Sco X-1 [90,97,98]. Here we search for GWs from young
74 non-pulsating neutron stars. The first search for these sources was done on the supernova remnant (SNR) Cas A [99]. We search for GWs from SNRs G65.7 and G330.2. The potential sources of interest for these directed searches were found by looking through Green’s SNR catalogue [100, 101]. The catalogue is filtered for Pulsar Wind Nebulae (PWNe) and Compact Central Objects (CCOs), and any sources with a well defined pulsation period are ruled out. The final filter is to pick X-ray sources with angular size < 3" , effectively a point source. The work done to select these sources is further detailed in [102]. The searches described here focus on young neutron stars found in SNRs because young NS are likely to be good generators of continuous GW signals [26]. Younger stars will be rotating more rapidly and thus generate a larger restoring force for the r-mode oscillations [103]. Typically the SNRs that have been searched over thus far have only been a few kyr old [104]. Due to the use of the resampling algorithm we can search G65.7 which is 20 kyr. G330.2 is 1 kyr old and is aligned more with the typical source for these searches. In these directed searches we aim to search over the largest ranges of frequency and spin-down parameters as possible. We model the frequency in the solar center barycenter frame for an object as
1 f ∼ f + f˙(t − t ) + f¨(t − t ) (4.1) 0 2 0 where t0 is the start of the observation time. We do not consider higher order terms because of the time spans and range of f and f˙ covered. This frequency expansion assumes that there were no glitches, that is, abrupt changes in the frequency of the star during our Tspan. The algorithm for setting up the “template bank” to efficiently cover the parameter space of f, f,˙ f¨, so that the minimal number of points is used to cover the largest amount of the space is described in [76] and Sec. 3.4.2. Looking back at Eq. 3.41 we see that the computational cost of the search ˙ ¨ 3 over (f, f, f) will scale as fmax. Therefore the range of frequencies to be searched over is bound by the computational cost. Because we do not know the spin-down parameters for these sources we estimate the range in f˙ using the range of f, the age of the SNR, and the range in f¨ is estimated from the range of f˙, f, and braking
75 index. Therefore we seek the largest possible range in f so that the ranges of f˙, and f¨ will be as large as possible. In order to place physically interesting upper limits (beating the indirect limit from energy conservation) on the sources we restrict ourselves to sources with at least a 100 Hz wide frequency band. The search is conducted using the technique of matched filtering see [30, 105], and Sec. 3.2 and Sec. 3.3 for more details. This technique takes advantage of a theoretical model for the GW signal we are hoping to find. As mentioned in Ch. 1 the idea is to compute the following integral
2 Z Tspan/2 (x||h) = x(t)h(t)dt (4.2) Sh(f0) Tspan/2
where x(t) represents the data stream, h(t) is the modeled signal, and Tspan is the amount of data used. This integral is used to create the F-statistic, which tests the null hypothesis that the input data contains no GWs. This integral must be computed numerically. The difficulty lies in the fact that in order to completely cover the space of spin-down parameters for the range of frequency we are interested in, one typical has 1013 models (templates) that must be integrated with the input data. This is where the large computational cost of the directed searches comes from. In order to reduce the computational cost we implemented the resampling algorithm (see Sec. 3.5 and [81]) in the search pipeline used for Cas A search [99]. This algorithm relies on the physical approximation that the Doppler modu- lation of an incoming signal due to the Earth’s rotation is small when compared to the Earth’s modulation due to the movement around the Sun. This allows the integral shown above to be computed using FFTs rather than the Dirchlet Kernel method. Although the speed up we achieved from resampling is not as great as that quoted in the original paper [81], it did allow us to examine G65.7 and G330.2. Without the speed up from resampling it would have been impossible to run a directed search like the one done for Cas A on these two objects. Indeed, the main usage for resampling for us was not to do the same computations faster but rather use the speed up to be able to search over objects which could not previously be searched over. In Sec. 4.2 of this chapter we give the details of running the search pipeline for these two sources. In Sec. 4.3 we discuss the implementation of the pipeline on the super-computing clusters at the Albert Einstein Institute, focusing on improvements
76 made to automate the pipeline. Having found no credible GW signal, we report upper limits for the strain, ellipticity and r-mode amplitude in Sec. 4.4. Finally, in Sec. 4.5 we discuss how our upper limits compare with previous LIGO searches and theoretical calculations.
4.2 Data analysis method
The search pipeline for directed searches for GWs from young non-pulsating neutron stars is summarized in Figure 4.1 and 4.2. We will use the figures as a guide for how a directed search is conducted. The first step is to install the prerequisite software, which includes the LALsuite software package (LAL), and Perl 5. LALsuite is a software package containing C and C++ code which is responsible for most of the computing power in the search, and is documented and maintained by the LVC. For the searches mentioned in this work we used version 6.7.0.1 of this software. Perl 5 is installed because many of the scripts which glue together the various LAL-code are written in Perl. In step two we find the astrophysical sources for the directed search. A list of potential sources of interest was already compiled, see discussion in Sec. 4.1. In order to find out which one of the targets on this list might provide a GW signal several estimates are made regarding the age-based GW limit and the computational cost of the target. The age-based limit is similar to the spin-down-limit used in various LIGO papers [55, 86]. We summarize the spin-down limit argument and then show how the age-based limit can be derived from it. The spin-down limit represents the GW strain that one would measure from an object if all of its rotational kinetic energy was converted to gravitational radiation. That is
dE ! 32G d 1 dE ! = I2 2(πf)6 ≤ − π2I f 2 = − , (4.3) dt 5c5 zz dt 2 zz dt GW KErot where f is the frequency of the source, Izz is the moment of inertia, and is the ellipticity. Using this we can solve for an upper limit on the ellipticity
77 Figure 4.1: This figure shows the first 12 steps of the data analysis pipeline used in directed gravitational wave searches. It also shows the iterations that are done at step 2, i.e. the high and low mini-searches. This figure was made by Dr. Ra Inta for the internal LIGO Scientific Collaboration (LSC) review of reference [104].
v u u 5c5 −f˙ t ≤ 4 5 . (4.4) 32π GIzz f 78 Figure 4.2: This figure shows steps 13 to 17 of the data analysis pipeline used in directed gravitational wave searches. This figure was made by Dr. Ra Inta for the internal LIGO Scientific Collaboration (LSC) review of reference [104].
Making use of the GW strain expected from such a source
4π2G I f 2 h = zz (4.5) 0 c4 D and our equation for the ellipticity, we get the upper limit on the strain amplitude
v u 1 u5GI −f˙ h ≤ t zz . (4.6) 0 D 2c3 f
In reality the kinetic energy is also lost to electromagnetic radiation, heating, etc., and so the spin-down limit is an optimistic upper-limit on the GW luminosity. Therefore, in order for a gravitational wave search to be meaningful the search must be able to find a GW strain that is lower than the spin-down-limit. The issue with using the spin-down limit for directed searches is that we
79 do not know the spin-frequency or its derivatives. Therefore, we can not use the spin-down limit as a way to select potential candidates. This is why we use an age-based limit. In order to transform the equations for the spin-down limit into one for the age-based limit we make use of the concept of the braking index n defined as ff¨ n = . (4.7) f 2 Then the range of f˙ is given by
f f − ≤ f˙ ≤ − , (4.8) (nmin − 1)a (nmax − 1)a and range of f¨ is given by
n f˙2 n f˙2 min ≤ f¨ ≤ max . (4.9) f f
For our searches we used nmin = 2 and nmax ranged from 3 to 7. If the spin-down is dominated by mass quadrupolar radiation then nmax = 5, and
s 5c5 age ≤ 4 4 (4.10) 128π GIzzτf s 1 5GI h ≤ zz . (4.11) age D 8c3τ
Using the numbers from the original methods paper [79] for the CCO Cas A we get
v !2 u 45 2 ! −4 100 Hz u 10 g cm 300 years age ≤ 3.9 × 10 t (4.12) f Izz τ v ! u ! 3.4 kpc u Izz 300 years h ≤ 1.2 × 10−24 t . (4.13) age D 1045 g cm2 τ
To determine whether a search is worthwhile the age-based-limit in Eq. 4.13 must be larger than the 95% confidence limit on the strain sensitivity of the detector which is given by s 95% Sh h0 = Θ , (4.14) Tdata where Θ is a statistical factor, typically in the low 30s, Sh is the one-sided power-
80 spectral density, and Tdata is the amount of data that is integrated over. One may think that any search is possible as long as one can take a great enough Tdata. There are several reasons why this does not work. First, there may simply not be enough data to integrate. Second, the increase in sensitivity only √ 7 goes as Tdata whereas the computational cost can scale as Tspan. Where
duty cycle = Tspan/Tdata (4.15) and the duty cycle represents the amount of data that remains after a suite of data quality vetoes have been applied. For our searches the duty cycle is typically 70%.
That is to say for the search over G330.2 where we used Tspan = 24 days, the Tdata was approximately 34 days. This means that while interferometers collected 34 days only 24 days of high quality data averaged over the two interferometers could be extracted. This is why in the search discussion that follows we speak of picking a Tspan because we can control how long of a data set we want, but we cannot control the quality of that data. ˙ ¨ The main input parameters for our search are f, f, f and the Tspan. We can equate Eq. 4.13 and Eq. 4.14 to get an estimate of the Tspan. In order to estimate the range of f, from which we can get f˙ and f¨, we need to estimate the computational 7 3 3 cost of the search. From Eq. 3.41 we know the cost will scale as Tspanf a . Where the a represents the age of the star and it is included because we used it to get the ranges of f,˙ f¨. From tests of the analysis pipeline it was discovered that the actual scaling 4 2.2 1.1 for the old version of Compute_FStatistic (CFS) was Tspanf a . The original search [99] which used this version was done in 2008 using the entire Atlas cluster and the computational cost was fixed at 3.5 days. The cost scaling for that search was f !2.2 T !4 300yr 1.1 3.5 days max span . (4.16) 300Hz 12days a When the resampling version of the CFS is used the scaling is approximately 1 4 2.2 1.1 3 Tspanf a . That is, the computational cost of our searches was fixed at 10.5 days of old-cluster equivalent time. By assuming a 70% duty cycle and equating Eq. 4.14 and 4.13 ,while keeping the computational cost fixed at 10.5 days we can solve for Tspan, fmin, fmax for each astrophysical object. This is done iteratively using a Python script. The script starts by taking fmax to be the most sensitive
81 frequency in the LIGO noise band. This is otherwise known as fbucket. Then the script essentially connects a line from fbucket to a lower frequency and that is fmin. Holding these two values fixed it solves for Tspan. Then using these three numbers, Tspan, fmin, fmax it calculates a computational cost using the empirical scaling Eq. 4.16 with 10.5 days instead of 3.5 days. If the cost is less than 10.5 days the script continues to increasing fmax, and repeats for 1 Hz bands until the cost is equal to or greater than 10.5 days. The output of this script is a directory with with the name of the source, with three sub-directories, low, high, and full, for each source. Each sub-directory contains a search_setup.xml file which is used by later steps in the pipeline. After this step a second iteration is done before the full search. The idea is to search around 10 Hz intervals around fmin and fmax. These mini-searches are done in the low and high subdirectories. There are several reasons to do this iteration. The first is to get a better estimate of the statistical factor Θ. This iteration also gives a more accurate value for fmin, fmax, and Tspan. The first iteration was done using PSD of the detectors harmonically averaged over the S6 run, but in the process of doing a mini search we get at the power spectrum density (PSD) of the detectors specifically for the optimal stretch of data of this source. Having this updated value for fmin and fmax we check that fmax − fmin is at least 100 Hz. After this second iteration we can also verify that higher order spin-derivatives will not be needed. This is done by using the parameter space metric [78] and computing the diagonal metric component for the third frequency derivative and verifying that the 2F lost by ignoring that term was less than the 20% maximum mismatch, see Sec. 3.4. After this first and second iteration were finished we found two sources which have not been previously searched over, and are only now possible due to the speed up from the resampling version of compute_FStatistic_v2. The sources are: G65.7+1.2. This source is also know as DA 495. The position and presence of a neutron star in this supernova remnant has been confirmed by Chandra observations [106]. The distance was estimated by [107–109]. It should be noted that in reference [107] the SNR is mislabeled as G55.7+1.2. The age was estimated by [109]. G330.2+1.0. The position of the SNR was measured by Chandra and XMM- Newton observations, see [110]. The distance was estimated by [111]. The age was
82 Table 4.1: Search targets and astronomical parameters
SNR Other RA+dec (J2000) D a G name name (hhmmss.s+ddmmss) (kpc) (kyr) 65.7+01.2 DA 495 19h 52m 10s + 15◦ 02’ 00" 8.5 0.54 330.2+01.0 16h 01m 06s −51◦ 34’ 00" 5 1 Target objects and astronomical parameters used in each search. Values of distance D and age a are at the optimistic (nearby and young) end of ranges given in the literature.
Table 4.2: Target objects, and astronomical parameters used in each search along with information about the SFTs used in the searches.
SNR fmin fmax Span Start of span H1 L1 Duty (G name) (Hz) (Hz) (days) (UTC, 2010) SFTs SFTs factor 65.7+01.2 136 263 51.63 Aug 14 00:53:35 1641 1666 0.67 330.2+01.0 146 254 24.36 Aug 14 00:53:35 756 897 0.70
estimated by [110] and [112]. The names, positions, estimated age, and distance of each source is summarized in Table 4.1. When possible we will highlight features of the pipeline using these two sources as examples. In step 3 of the pipeline, directories (“global”, and “local”) are set up in each sub-directory, which stores the data used in the search. The LVC uses data in the form of Short-Fourier-Transforms or SFTs. These data products come from taking 30 minutes of time-series strain data as measured by the interferometers, high-pass filtering them at 40Hz, passing them through a Tukey window and then taking the Fourier transform.
4.2.1 Data Prepration
In step 4 the locations for the SFTs from the science run from July 2009 to October 2010, henceforth known as S6 are collected, and stored in a database file, sft_database.xml.bz2. The locations are gathered using a Python script in LAL called ligo_data_find. We used this script to get data from the interferometers at Livingston (L1) and Hanford (H1). The data from the Virgo interferometer is not used. Although V1 is more sensitive at low frequencies, around 40 Hz, the
83 searches done here were started around 100 Hz. And at these higher frequencies the sensitivity of L1 and H1 are a order of magnitude better than V1. In step 5 an average PSD is calculated for each SFT in the frequency range of the search, using lalapps_computePSD. In step 6 the results from this calculation are added to the database created in step four. In step 7 this stored information is
used to find the optimal stretch of data for the Tspan determined in step 2. The selection method used here is the same as the one used in [113]:
1 X (4.17) f,k Sh(f, t)
Here f is the frequency in each bin, t is the time stamp of the SFT, and Sh(f, t) is the strain noise PSD calculated in step 5. Maximizing this figure of merit roughly corresponds to looking for the most sensitive stretch of data. The sum is dominated by least noisy frequencies and thus the times selected were near the end of S6 data.
For example the Tspan starts at Aug 14 2010 for both G65.7 and G330.2.
4.2.2 Search
In step 8 the density of templates for a 1 Hz frequency band centered at 100 Hz is calculated. In step 9 these densities are stored into template_density.xml. In step 10 we calculated the cost per template of running a search for a 1 Hz frequency band centered around 100 Hz. Coupled with the knowledge of template density we can estimate what the cost of a frequency band will be. In step 11 the frequency bands are set up, using the cost per band calculated in step 10, such that the search over each band takes approximately the same amount of time. Along with this calculation step 11 also copies over the SFTs (which were found to have the optimal signal to noise ratio in step 7) from the LIGO servers into the directories set up in step 3. Step 12 calculates the multi-inteferometer F-statistic [30] for each template in the parameter space. From the discussion in Sec. 3.3 we know the F-statistic will be drawn from a χ2 distribution. Using this we set a threshold value of 33 below which the result from the F-statistic calculation is not stored. This roughly corresponds to keeping 1 in a million values of 2F. Doing this helps to create result files of a more manageable size.
84 4.2.3 Validation
Before manually investigating a detection candidate two more automated vetoes are used. Both of these vetoes are done in Step 13. The first is the Fscan veto. This veto eliminates spurious signals that arise from non-stationary noise. The veto uses the following procedure. First, create a spectrogram from the SFTs use in the search, and normalize the power by using a running median of 50 frequency bins. Then the power in these bins is time averaged. If the noise were stationary and Gaussian the power thus calculated would be drawn from a χ2 distribution. Therefore, if we find frequency bins with large deviations from the χ2 distribution then we know that bin has non-stationary noise. For the searches described here the threshold was set to ±7 deviations. The next veto is the interferometer consistency veto. This veto uses the fact that the combined value of 2F for an individual template, calculated using data from both interferometers, will be larger than the 2F calculated using data from one just one interferometer. Templates that violate this rule are eliminated. The reader may have noticed that this is a veto on individual templates whereas the Fscan vetoed entire search bands. In order to put the two vetoes on more equal footing the consistency veto is extended to full bands by the following argument. If the number of vetoed templates in a frequency band is larger than the number of non-vetoed templates then the entire band is vetoed. These vetoes have been tested in [113] and [104] and they are deemed safe, in the sense that they do not eliminate hardware injected signals. These are signals are created in the LIGO data stream by literally moving the test mirrors. Finally, step 13 also calculates the full PSD for the specific source, given the
frequency range and Tspan. The PSD information is used for setting upper limits. After the vetoing is complete we can run step 14 of the pipeline. This step processes the search jobs which have passed all the automatic vetoes and collects it into search_bands.xml and upper_limit_bands.xml. These XML files along with a file indicating which jobs got vetoed is accessed using a Mathematica notebook (post.nb), which is run on a local computer not the computing clusters. One of the first things done in post.nb is to examine the frequency bands that were vetoed. The plot of the bands vetoed for G65.7 is shown in Fig. 4.3. We notice that there are bands vetoed at and around 180 Hz, and 240 Hz. This make
85 1.0
0.8
0.6
0.4
0.2
0.0 140 160 180 200 220 240 260
Figure 4.3: This figure shows the frequency bands that were vetoed for the search over G65.7. This is a graphical representation of the vetoes took place in step 13 of the data analysis pipeline. sense since they are harmonics of the 60 Hz power-lines. Some of the other veto bands are digital electronics lines, including many 2 Hz harmonics. Next we examine the templates with the greatest value for the F-statistic which survived all the vetoes mentioned thus far. We denote the largest value of the statistic as 2F ?. The probability that a given value of 2F ? will be observed when no signal is present is given as
"Z 2F ? #N−1 ? 2 ? 2 p(2F ) = Np(χ4; 2F ) d(2F)p(χ4; 2F) , (4.18) 0 where N represent the number of templates, and p(2F ?) denotes the probability density of a χ2 distribution with four degrees of freedom. If 2F ? is such that p is greater than 5% then that template is not significant and ruled out as a potential candidate. However, if 2F ? is unlikely, i.e. has probability of less than 5% of being drawn from the χ2 distribution randomly then that template is worth further investigation. This sets a threshold value for 2F ? above which we keep templates.
86 The distribution of 2F ? values for G65.7 over the search band is shown in Fig. 4.4. loudest nonvetoed 2F
65. 60.
55.
50.
45.
frequency(Hz) 160 180 200 220 240 260
Figure 4.4: This figure show the distribution of the loudest 2F values that survived the vetoing procedure that takes place in step 13.
Equation 4.18 above assumes the N are all statistically independent. This is not true because the 2F ? value of neighboring candidates (templates) is correlated. For the searches done in [113] it was seen that the effective number of templates was approximately 90% of the total number of templates. Therefore the value set for threshold is not affected greatly. In our searches the use of resampling greatly affected the effective number of templates and we saw the effective number go down to 50%. The reason for this is still being investigated. This reduced number of effective templates lowered the threshold value for 2F ? as compared to what it would have been with a higher number of effective templates but the thresholding procedure remains the same as [113]. The jobs that are left over after the 2F ? threshold veto are manually investi- gated by comparing the theoretical distribution of 2F ? values for that job versus the one from the data. After this, a text file called “man_v_jobs.txt” is uploaded to the cluster, with the list of jobs that were manually vetoed. It is not shown in the flowchart but in the most recent version of the pipeline we iterate this manual veto process. That is, after uploading the “man_v_jobs.txt” file we run step 14 again. Then run post.nb to see if any jobs survived. When we did this for our sources we did not find any remaining jobs and therefore we do not claim a GW
87 detection.
4.2.4 Upper Limits
As with all LVC searches up to date our search for G65.7 and G330.2 did not yield a detection of GWs. Therefore, we place upper limits on GW strain, the ellipticity, and the r-mode amplitude, that can be expected from these sources. This starts with running step 15 of the pipeline. We look for the loudest 2F ? in each 1 Hz ? frequency band. Then we determine the h0 that would lead to such a value of 2F using an analytical calculation and a computationally inexpensive Monte Carlo simulation [113]. After this step a computationally more expensive step used to ? verify that this value of h0 indeed gives the 2F reported is undertaken. This is done in step 16 by injecting fake GW signals across each 1 Hz frequency band. In [113] 6000 fake signals were used for each frequency band. The signals all ˙ ¨ shared the same h0 but differ in values of f, f, f, and extrinsic parameters. Due to increased computational cost when using the resampling algorithm for this step, we used only 600 injections per frequency bin and we did not examine every 1 Hz bin but rather injected signals at 5 Hz intervals. In Sec. 4.5 we discuss this issue further. In step 17 the results from these injections are collated and stored in upper_limit_bands.xml. The final step of the pipeline is to run post.nb again on the local computer. If post.nb is run after completing a high and low mini-search the notebook is used to get better estimates of Θ, fmin, fmax, and Tspan. If post.nb is run after the full search, it is used to examine the upper limits and make the plots shown in Sec. 4.4.
4.3 Implementation
Due to the vast amount of data that must be processed in doing one of these searches, we must make use of various computational resources. These searches are run on the Atlas cluster at AEI (The Max Planck Institute for Gravitational Physics) in Hannover, Germany. After a recent upgrade there are now approximately 10,000 nodes. The newly installed nodes are also faster since they make use of various new technologies including GPU-processing. The mixture of new and old nodes complicated our attempts to estimate the computational cost scaling. The new
88 nodes seemed to be faster by about a factor of two. These computing clusters make use of a batch submission program called High Throughput Condor (HTC). The main idea behind HTC is that the user creates a submit file (?.sub) with specifications about how to run a particular process on the cluster. These include, but are not limited to, the name of the executable to be launched, arguments to the executable, where to place output, error, and log files, what kind of computer processor to use, etc. Once the user issues the command condor_submit ?.sub, HTC tries to match the description in the ?.sub file to the available computational resources. For the search over SNR G330.2 we used approximately 1964 jobs and for G65.7 we used approximately 2764. In the Cas A paper [113] the cost was about 30,000 jobs. The discrepancy is easy to understand when one accounts for the fact that we gained a 3 times speed up from resampling and then a 2 times speed up from the cluster upgrades. The remaining factor of 4 comes from the fact that the individual search jobs had larger frequency bands in our searches than in the Cas A search. One of the shortcomings of using condor jobs is that they cannot be scheduled. When the command condor_submit is executed for files, A.sub and B.sub, HTC will start running them simultaneously, even if the output of A.sub needs to be used as input to B.sub. What has been done in the past is simply to wait for the process to be done in A.sub and then condor_submit B.sub; however, this can be quite inefficient. The solution is to use a scheduling program which will submit the jobs in the necessary order. The scheduler which comes with HTC is called DAGman, which stands for Directed Acyclic Graph Manager. A DAG file can be used to code in dependencies between condor jobs. The search pipeline described in Sec. 4.2 and used in [113] only used condor jobs, we have since improved the pipeline by using DAGs. All the steps of the pipeline already discussed are still used. The main idea is that we have essentially programmed in the flowchart in Fig. 4.1 and 4.2 by using the DAG structure depicted in Figure 4.5.
When running the low and high searches to get better estimates of Θ, fmin, fmax, and Tspan, we ran the Master DAG. This is done by the command con- dor_submit_dag Master.dag. This DAG runs the search pipeline from step 3 to step 17, as described in Sec. 4.2. When running a full search we first ran SetUp.dag and Search.dag which calls sub-dag Steps3-11 which in turn runs those steps of
89 Master
SetUp and Upper Search Limits
Steps Steps Steps 3-11 post.nb Step 15 post.nb 12-14 16-17
XML files XML files
Figure 4.5: This figure is a schematic of the Directed Acyclic Graph we used to automate the data analysis pipeline used for directed searches. the pipeline. After this sub-dag is finished, the Steps 12-14 sub-dag was called and then we manually processed the vetoed jobs in the mathematica notebook post.nb. When the manual vetoing was finished and after the man_v_jobs.txt file was uploaded we ran UpperLimits.dag. This ran the two sub-dags, Steps15.dag and Steps16-17.dag. The sub-dags are necessary because the output of step 15 results in a variable number of jobs, which is not known until that DAG is completed. Once the number of jobs is known then Steps16-17.dag can be run. Finally we ran post.nb again to analyze the results of the search, and created upper limit plots. In the future it may be possible to run the Master DAG for the full search. The key element that will have to be changed is post.nb. This will possibly require re-writing the contents of the notebook into a Python or Perl script, and automating the manual vetoes, neither of which are trivial tasks.
4.4 Results
In this section we present the results from our searches for continuous gravitational waves from supernova remnants G65.7 and G330.2. Having found no statistically
90 significant GW signals we place upper limits on the gravitational wave strain, the ellipticity, and the r-mode amplitude of these sources.
The strain amplitude (h0) represents the magnitude of the GW. Here we plot the values of h0 in 1 Hz band for each search. The shape of the strain graph closely resembles the LIGO PSD graph shown in Fig. 1.8. This is difficult to see in our plots because the frequency band presented is much smaller than that shown in Fig. 1.8. It is not surprising then, that the best upper limits on strain amplitude occur between 150–180 Hz since this is where the detectors are most sensitive. The upper limits on strain can be translated into upper limits on the fiducial ellipticity defined in Eq. 4.13. Starting with the age-based limit for ellipticity and 45 2 substituting a fiducial value of Izz = 10 g cm we get
! ! 1 !2 h a 2 100 Hz = 3.9 × 10−4 0 . (4.19) age 1 × 10−24 300 yr f
We use this equation to convert the age-based limit of h0 and the 95% confidence limits obtained here. It should be noted that this ellipticity is a dimensionless version of the non-axisymmetric part of the mass quadrupole moment, not the actual shape of the star. The quantity inferred from h0 is really a component of the mass quadrupole moment. The conversion factors to the these quantities can be found in [114] and are known to have uncertainties of a factor of 5 depending on the star’s mass and equation of state. The strain upper limits can also be translated to the upper limits on the r-mode amplitude, α as shown by [115], via
h ! 100Hz ! D ! α = .28 0 . (4.20) 10−24 f 1kpc
For a typical NS there is an uncertainty of 2-3 in α depending on the star’s mass and equation of state. Just as in the case of ellipticity Eq. 4.20 is used to convert the age-based limit of h0 and the 95% confidence limits obtained here. Similar to the case of the fiducial ellipticity the quantity inferred from α is a component of the current quadrupole moment, and the conversion factor can be uncertain by a factor of a few [103]. In Fig. 4.6 we plot strain amplitude and the upper limits for G65.7. The dots represent the 95% GW strain for that frequency and the straight black line is the
91 indirect limit on the strain set by the age of the star (see Eq. 4.13). The most sensitive strain measure for this source was about 2.9 × 10−25 which occurred at 169 Hz. In figure 4.7 we plot the fiducial ellipticity of G65.7 using Eq. 4.19. The
�� ��� ���������(��) ��� �����
6.×10 -25
5.×10 -25
4.×10 -25
3.×10 -25
2.×10 -25
1.×10 -25
140 160 180 200 220 240 260
Figure 4.6: This figure shows the strain amplitude upper limit for G65.7 (dark circles) and the indirect age-based upper limit (solid line). ellipticity ranged from 2.6 × 10−5 to 7.0 × 10−6. In Fig. 4.8 we plot the r-mode amplitude using Eq. 4.20. The values of α ranged from 2.2 × 10−4–1.6 × 10−3. In Fig. 4.9 we plot the the upper limits for G330.2. The most sensitive strain measure for this source was about 4 × 10−25 which occurred at 163 Hz. The −5 −5 ellipticity inferred from h0 ranged from 3.7 × 10 –9.9 × 10 and is plotted in −3 −3 Fig. 4.10. The r-mode amplitude inferred from h0 ranged from 1.7×10 –7.4×10 and is plotted in Fig. 4.11.
4.5 Discussion
The most sensitivity value of h0(t) found in these searches came from G65.7 at approximately 3 × 10−25, at 169 Hz. The LIGO PSD achieves the best signal to noise ratio at 170 Hz. Thus it is reasonable that our strongest (lowest) upper limit
92 ϵ ��� ���������(��) ��� ����� 3.0×10 -5
2.5×10 -5
2.0×10 -5
1.5×10 -5
1.0×10 -5
160 180 200 220 240 260
Figure 4.7: This figure shows the ellipticity upper limit for G65.7 inferred from the strain amplitude (dark circles) and the indirect age-based upper limit (solid line). occurred close to this frequency. This upper limit is twice as sensitive as that presented in the original Cas A paper. There are several reasons why we were able to gain a greater sensitivity. For one, the LIGO instrument itself was more sensitive. The improvements from S5 data, on which the Cas A search was done to the S6 data, used here, is approximately 30%. Second, by making use of resampling and a smaller frequency band, 127 Hz versus 200 Hz for the Cas A search [113], we were able to integrate much more data. The Cas A search integrated 12 days of data, here we integrated 51 days. Roughly speaking this explains the factor of 2 increase, since the sensitivity goes as the square root of Tspan, and we used a little over 4 times as much data. Let us compare our numbers for ellipticity and r-mode amplitudes with the estimates in the literature. For G65.7 our range for ellipticity is 7.0 × 10−6– 2.6 × 10−5, and beat the maximum fiducial ellipticity 10−5 for normal NS [116], and the maximum fiducial ellipticity of 10−3 for quark based NS [116] over the whole search band. For G330.2 our range for ellipticity is 9.9 × 10−5–3.7 × 10−5 and this only beat maximum fiducial ellipticity 10−5 for normal NS at the high end of the
93 α ��� ���������(��) ��� ����� 0.002
0.001
5.×10 -4
2.×10 -4 160 180 200 220 240 260
Figure 4.8: This figure shows the r-mode upper limit for G65.7 inferred from the strain amplitude (dark circles) and the indirect age-based upper limit (solid line).
h0 vs. frequency HHzL for G330.2
8. ´ 10-25
6. ´ 10-25
4. ´ 10-25
2. ´ 10-25
160 180 200 220 240 Figure 4.9: This figure shows the strain amplitude upper limit for G330.2 (dark circles) and the indirect age-based upper limit (solid line).
94 ϵ ��� ���������(��) ��� ������
1.×10 -4
8.×10 -5
6.×10 -5
4.×10 -5
160 180 200 220 240
Figure 4.10: This figure shows the ellipticity upper limit for G330.2 inferred from the strain amplitude (dark circles) and the indirect age-based upper limit (solid line). search band. Our range of r-mode amplitudes for G65.7 is 2.2 × 10−4–1.6 × 10−3 and for G330.2 it is 1.7 × 10−3–7.4 × 10−3. These values are close to beating the theoretical limit of 10−3 [117]. The use of the resampling algorithm did make it possible to search for GW from these sources. However things are far from perfect. We see from the upper limit plots that the search barely beat the age-based-limit. This is because our cost scaling was off. The scaling was thrown off because the speed of the resampling algorithm depends on the frequency band of the search, and this was not taken into account by the method used in step 10 of the pipeline, which estimates the cost of computing the F-statistic for a template centered around 100 Hz. If we had a better estimate of the cost scaling we would have been able to place stronger upper limits on our searches. Along with the cost scaling issue, resampling did also prevent us from using the same method as [113] for the upper limit injections. The main issue is that the
95 α ��� ���������(��) ��� ������
0.008
0.006
0.004
0.002
160 180 200 220 240
Figure 4.11: This figure shows the r-mode upper limit for G330.2 inferred from the strain amplitude (dark circles) and the indirect age-based upper limit (solid line). injections were taking much longer when using CFS-resamp, versus the older CFS. The reason for this is because the injections use very narrow-band signals in f, f,˙ f¨, and this is an issue for resampling because of the interpolation step in the algorithm. In order to work around the issues of interpolation errors the current algorithm just doubles the frequency band they are searching. However, for injections this creates a lot of overhead and simply creates too large of a computational cost. Therefore we used only 600 injections per frequency bin and we did not examine every 1 Hz but rather injected signals at 5 Hz intervals. One of the improvements to the search pipeline would be to improve the interpolation scheme used in resampling. This would make it possible to conduct more UL injections, and it may help with the issues related to cost scaling. If nothing else, it would bring down the computational cost of running the upper limits. Even with these issues we were able to use resampling for its main purpose: to speed up computation of the F-statistic. This allowed us to conduct searches on SNRs G65.7 and G330.2, which would not have been possible without the speed up from resampling. Furthermore, for G65.7 we were able to set strong upper
96 limits, essentially twice the sensitivity of the original Cas A search. Given that the advanced detectors will be able to achieve instantaneous strains of 10−25 we might hope that using the resampling algorithm on that data will allow us to set even stronger upper limits for strain amplitude, possibly as low as 10−26.
97 Chapter 5 | Conclusion
In this dissertation we presented research on gravitational wave searches from neutron star sources. In Ch. 1 we provided an overview of GWs, their sources, propagation, and energy carried. We also introduced NS and their r-mode oscillations. Finally, we looked at the LIGO detectors and discussed the data analysis challenge of detecting a GW signal. In Ch. 3 we extended the discussion of GW data analysis. This involved a further look into the technique of matched filtering, a discussion of the detection statistic, a discussion of the parameter space used for directed GW searches, and the resampling algorithm used to reduce the computational cost of a search. In Ch. 2 we presented new research into the r-mode oscillations of NS. Our main goal was to examine how the r-mode frequency was changed when realistic equations of state were used to model the star, as opposed to the standard polytropic models used previously in this research field. In order to determine the nature of this change we calculated values for a dimensionless measure of the r-mode frequency, κ, for 14 different equations of state over their equilibrium sequence. Our results define a narrow frequency band over which to search for GWs from r-modes. These results can be applied to narrow-band searches for GWs of the type discussed in [56], which claimed to achieve a sensitivity of twice that of the original narrow-band search conducted on the Crab pulsar [85]. If researchers combine our results, the narrow-band search techniques of [56] and perhaps the resampling algorithm [81], then even greater sensitivity in GW searches can be achieved. In Ch. 4 we presented directed searches for GWs from non-pulsating neutron stars in supernova remnants. In these searches the sky-position of the source is
98 known very well, and one searches over the spin frequency and frequency derivatives of the sources. We were only able to search these two sources due to the speed up in computation of the detection statistic due to the resampling algorithm. Although there is more work to be done in efficiently applying this algorithm it does increase the number of viable GW sources, and this may lead to physically interesting results, such as those we presented in Ch. 4. Although our results for these searches did not yield a credible GW signal we did set strong upper limits on the GW strain amplitude. The upper limits achieved for G65.7+1.2 was twice as sensitive as those achieved by the Cas A search [113]. With the launch the of new detectors next year, new data analysis methods, and new research like that presented in this work, the sensitivity of GW searches may be dramatically improved. It seems that almost a hundred years after the prediction of gravitational waves the wait for a direct detection may be coming to an end.
99 Bibliography
[1] Pitkin, M., S. Reid, S. Rowan, and J. Hough (2011) “Gravitational Wave Detection by Interferometry (Ground and Space),” Living Reviews in Relativity, 14, p. 5, 1102.3355.
[2] Smith, M., D. Fox, D. Cowen, P. MÃľszÃąros, G. TeÅąiÄĞ, J. Fixelle, I. Bartos, P. Sommers, A. Ashtekar, G. J. Babu, S. Barthelmy, S. Coutu, T. DeYoung, A. Falcone, S. Gao, B. Hashemi, A. Homeier, S. MÃąrka, B. Owen, and I. Taboada (2013) “The Astrophysical Multimessenger Observatory Network (AMON),” Astroparticle Physics, 45(0), pp. 56 – 70. URL http://www.sciencedirect.com/science/article/pii/ S0927650513000479
[3] Hulse, R. A. and J. H. Taylor (1975) “Discovery of a pulsar in a binary system,” Astrophys. J. l, 195, pp. L51–L53.
[4] Weisberg, J. M., D. J. Nice, and J. H. Taylor (2010) “Timing Mea- surements of the Relativistic Binary Pulsar PSR B1913+16,” Astrophys. J., 722(2), p. 1030. URL http://stacks.iop.org/0004-637X/722/i=2/a=1030
[5] Ade, P. A. R. and Bicep2 Collaboration (2014) “Detection of B-Mode Polarization at Degree Angular Scales by BICEP2,” Physical Review Letters, 112(24), 241101, 1403.3985.
[6] Planck Collaboration, R. Adam, P. A. R. Ade, N. Aghanim, M. Ar- naud, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, J. G. Bartlett, and et al. (2014) “Planck intermediate results. XXX. The angular power spectrum of polarized dust emission at intermediate and high Galactic latitudes,” ArXiv e-prints, 1409.5738.
[7] Baumgarte, T. and S. Shapiro (2010) Numerical Relativity: Solving Einstein’s Equations on the Computer, Cambridge University Press. URL http://books.google.com/books?id=ZPogAwAAQBAJ
100 [8] Cook, G. B. (2000) “Initial Data for Numerical Relativity,” Living Reviews in Relativity, 3(5). URL http://www.livingreviews.org/lrr-2000-5
[9] Gourgoulhon, E. (2007) “3+1 Formalism and Bases of Numerical Relativity,” ArXiv General Relativity and Quantum Cosmology e-prints, gr-qc/0703035.
[10] Patel, P. (2011) Search for gravitational waves from a nearby neutron star using barycentric resampling, Ph.D. thesis, California Institute of Technology.
[11] Thorne, K. S. (1980) “Multipole expansions of gravitational radiation,” Rev. Mod. Phys., 52, pp. 299–339. URL http://link.aps.org/doi/10.1103/RevModPhys.52.299
[12] Carroll, B. and D. Ostlie (2007) An Introduction to Modern Astrophysics, Pearson Addison-Wesley. URL http://books.google.com/books?id=M8wPAQAAMAAJ
[13] Glendenning, N. (2000) Compact Stars: Nuclear Physics, Particle Physics, and General Relativity, Astronomy and Astrophysics Library, Springer New York. URL http://books.google.com/books?id=BBSOgxe2S1AC
[14] Shapiro, S. and S. Teukolsky (2008) Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Objects, Wiley. URL http://books.google.com/books?id=d1CRQIcP1zoC
[15] Baade, W. and F. Zwicky (1934) “Remarks on Super-Novae and Cosmic Rays,” Physical Review, 46, pp. 76–77.
[16] Chadwick, J. (1932) “Possible existence of a neutron,” Nature, 129(3252), p. 312.
[17] Lattimer, J. M. and M. Prakash (2010) “What a Two Solar Mass Neutron Star Really Means,” ArXiv e-prints.
[18] Demorest, P. B., T. Pennucci, S. M. Ransom, M. S. E. Roberts, and J. W. T. Hessels (2010) “A two-solar-mass neutron star measured using Shapiro delay,” Nature, 467, pp. 1081–1083.
[19] Clayton, D. (1968) Principles of Stellar Evolution and Nucleosynthesis, University of Chicago Press. URL http://books.google.com/books?id=8HSGFThnbvkC
[20] Riles, K. (2013) “Gravitational waves: Sources, detectors and searches,” Progress in Particle and Nuclear Physics, 68, pp. 1–54, 1209.0667.
101 [21] Horowitz, C. J. and K. Kadau (2009) “Breaking Strain of Neutron Star Crust and Gravitational Waves,” Phys. Rev. Lett., 102, p. 191102. URL http://link.aps.org/doi/10.1103/PhysRevLett.102.191102
[22] Bildsten, L. (1998) “Gravitational Radiation and Rotation of Accreting Neutron Stars,” Astrophys. J. Lett., 501, pp. L89–L93.
[23] Lindblom, L. “Gravitational Radiation Instabilities in Rotating Neutron Stars,” in Proceedings of the Eleventh Workshop on General Relativity and Gravitation in Japan (e. a. Koga, J., ed.).
[24] Friedman, J. L. and S. M. Morsink (1998) “Axial Instability of Rotating Relativistic Stars,” Astrophys. J., 502(2), p. 714. URL http://stacks.iop.org/0004-637X/502/i=2/a=714
[25] Andersson, N. (1998) “A New Class of Unstable Modes of Rotating Rela- tivistic Stars,” Astrophys. J., 502, p. 708.
[26] Owen, B. J., L. Lindblom, C. Cutler, B. F. Schutz, A. Vecchio, and N. Andersson (1998) “Gravitational waves from hot young rapidly rotating neutron stars,” Phys. Rev. D, 58, p. 084020. URL http://link.aps.org/doi/10.1103/PhysRevD.58.084020
[27] Carroll, S. (2004) Spacetime and Geometry: An Introduction to General Relativity, Addison-Wesley Longman, San Francisco. URL http://books.google.com/books?id=1SKFQgAACAAJ
[28] Abbott, B. P., R. Abbott, R. Adhikari, P. Ajith, B. Allen, G. Allen, R. S. Amin, S. B. Anderson, W. G. Anderson, M. A. Arain, and et al. (2009) “LIGO: the Laser Interferometer Gravitational- Wave Observatory,” Reports on Progress in Physics, 72(7), 076901, 0711. 3041.
[29] Sathyaprakash, B. S. and B. F. Schutz (2009) “Physics, Astrophysics and Cosmology with Gravitational Waves,” Living Reviews in Relativity, 12, p. 2, 0903.0338.
[30] Jaranowski, P., A. Krolak, and B. F. Schutz (1998) “Data analysis of gravitational-wave signals from spinning neutron stars. I: The signal and its detection,” Phys. Rev. D, 58, p. 063001, gr-qc/9804014.
[31] Helstrom, C. (1960) Statistical Theory of Signal Detection, International series of monographs on electronics and instrumentation, Macmillan. URL http://books.google.com/books?id=RNAyAAAAMAAJ
102 [32] M Punturo, e. a. (2010) “The third generation of gravitational wave observatories and their science reach,” Classical and Quantum Gravity, 27(8), p. 084007. URL http://stacks.iop.org/0264-9381/27/i=8/a=084007
[33] S Hild, e. a. (2011) “Sensitivity studies for third-generation gravitational wave observatories,” Classical and Quantum Gravity, 28(9), p. 094013. URL http://stacks.iop.org/0264-9381/28/i=9/a=094013
[34] Somiya, K. (2012) “Detector configuration of KAGRAâ the Japanese cryo- genic gravitational-wave detector,” Classical and Quantum Gravity, 29(12), p. 124007. URL http://stacks.iop.org/0264-9381/29/i=12/a=124007
[35] M Armano, e. a. (2009) “LISA Pathfinder: the experiment and the route to LISA,” Classical and Quantum Gravity, 26(9), p. 094001. URL http://stacks.iop.org/0264-9381/26/i=9/a=094001
[36] Papaloizou, J. and J. E. Pringle (1978) “Non-radial oscillations of rotating stars and their relevance to the short-period oscillations of cataclysmic variables,” Mon. Not. R. Astron. Soc., 182, pp. 423–442.
[37] Chandrasekhar, S. (1970) “Solutions of Two Problems in the Theory of Gravitational Radiation,” Phys. Rev. Lett., 24, pp. 611–615. URL http://link.aps.org/doi/10.1103/PhysRevLett.24.611
[38] Friedman, J. L. and B. F. Schutz (1978) “Secular instability of rotating Newtonian stars,” Astrophys. J., 222, pp. 281–296.
[39] Lindblom, L., B. J. Owen, and S. M. Morsink (1998) “Gravitational Radiation Instability in Hot Young Neutron Stars,” Phys. Rev. Lett., 80, pp. 4843–4846.
[40] Andersson, N., K. Kokkotas, and B. F. Schutz (1999) “Gravitational Radiation Limit on the Spin of Young Neutron Stars,” Astrophys. J., 510, pp. 846–853.
[41] Andersson, N., K. D. Kokkotas, and N. Stergioulas (1999) “On the Relevance of the R-Mode Instability for Accreting Neutron Stars and White Dwarfs,” Astrophys. J., 516, pp. 307–314.
[42] Watts, A. L. (2012) “Thermonuclear Burst Oscillations,” Ann. Rev. of Astron. and Astrophys., 50, pp. 609–640.
[43] Strohmayer, T. and S. Mahmoodifar (2014) “A Non-radial Oscillation Mode in an Accreting Millisecond Pulsar?” Astrophys. J., 784, 72.
103 [44] ——— (2014) “Discovery of a Neutron Star Oscillation Mode During a Superburst,” Astrophys. J. Lett., 793, L38.
[45] Andersson, N., D. I. Jones, and W. C. G. Ho (2014) “Implications of an r mode in XTE J1751-305: mass, radius and spin evolution,” Mon. Not. R. Astron. Soc., 442, pp. 1786–1793.
[46] Lee, U. (2014) “Excitation of a non-radial mode in a millisecond X-ray pulsar XTE J1751-305,” Mon. Not. R. Astron. Soc., 442, pp. 3037–3043.
[47] Provost, J., G. Berthomieu, and A. Rocca (1981) “Low Frequency Oscillations of a Slowly Rotating Star - Quasi Toroidal Modes,” Astron. and Astrophys., 94, p. 126.
[48] Lindblom, L., G. Mendell, and B. J. Owen (1999) “Second-order rotational effects on the r-modes of neutron stars,” Phys. Rev. D, 60, p. 064006. URL http://link.aps.org/doi/10.1103/PhysRevD.60.064006
[49] Lockitch, K. H., J. L. Friedman, and N. Andersson (2003) “Rotational modes of relativistic stars: Numerical results,” Phys. Rev. D, 68, p. 124010. URL http://link.aps.org/doi/10.1103/PhysRevD.68.124010
[50] Stergioulas, N. and J. A. Font (2001) “Nonlinear r-Modes in Rapidly Rotating Relativistic Stars,” Phys. Rev. Lett., 86, pp. 1148–1151.
[51] Gaertig, E. and K. D. Kokkotas (2008) “Oscillations of rapidly rotating relativistic stars,” Phys. Rev. D, 78(6), 064063.
[52] Yoshida, S., S. Yoshida, and Y. Eriguchi (2005) “R-mode oscillations of rapidly rotating barotropic stars in general relativity: analysis by the relativistic Cowling approximation,” Mon. Not. R. Astron. Soc., 356, pp. 217–224.
[53] Read, J. S., B. D. Lackey, B. J. Owen, and J. L. Friedman (2009) “Constraints on a phenomenologically parametrized neutron-star equation of state,” Phys. Rev. D, 79(12), 124032.
[54] Lindblom, L. and N. M. Indik (2012) “A Spectral Approach to the Relativistic Inverse Stellar Structure Problem,” Phys. Rev. D, D86, p. 084003.
[55] Abbott, B. and et al. (2008) “Beating the Spin-Down Limit on Grav- itational Wave Emission from the Crab Pulsar,” Astrophys. J., 683, pp. L45–L49.
104 [56] Astone, P., A. Colla, S. D’Antonio, S. Frasca, C. Palomba, and R. Serafinelli (2014) “Method for narrow-band search of continuous gravitational wave signals,” Phys. Rev. D, 89(6), 062008.
[57] Lattimer, J. M. and M. Prakash (2007) “Neutron star observations: Prognosis for equation of state constraints,” Phys. Rep., 442, pp. 109–165.
[58] Manchester, R. N., G. B. Hobbs, A. Teoh, and M. Hobbs (2005) “The Australia Telescope National Facility Pulsar Catalogue,” Astron. J., 129, pp. 1993–2006.
[59] Bildsten, L. and G. Ushomirsky (2000) “Viscous Boundary-Layer Damp- ing of R-Modes in Neutron Stars,” Astrophys. J. Lett., 529, pp. L33–L36.
[60] Levin, Y. and G. Ushomirsky (2001) “Crust-core coupling and r-mode damping in neutron stars: a toy model,” Mon. Not. R. Astron. Soc., 324, pp. 917–922.
[61] Yoshida, S. and U. Lee (2000) “Rotational Modes of Nonisentropic Stars and the Gravitational Radiation-driven Instability,” Astrophys. J. Suppl., 129, pp. 353–366.
[62] Passamonti, A., B. Haskell, N. Andersson, D. I. Jones, and I. Hawke (2009) “Oscillations of rapidly rotating stratified neutron stars,” Mon. Not. R. Astron. Soc., 394, pp. 730–741.
[63] Ho, W. C. G. and D. Lai (2000) “r-Mode Oscillations and Spin-down of Young Rotating Magnetic Neutron Stars,” Astrophys. J., 543, pp. 386–394.
[64] Morsink, S. M. and V. Rezania (2002) “Normal Modes of Rotating Magnetic Stars,” Astrophys. J., 574, pp. 908–919.
[65] Lander, S. K., D. I. Jones, and A. Passamonti (2010) “Oscillations of rotating magnetized neutron stars with purely toroidal magnetic fields,” Mon. Not. R. Astron. Soc., 405, pp. 318–328.
[66] Lockitch, K. H., N. Andersson, and J. L. Friedman (2001) “Rotational modes of relativistic stars: Analytic results,” Phys. Rev. D, 63(2), 024019.
[67] Lindblom, L. (1992) “Determining the nuclear equation of state from neutron-star masses and radii,” Astrophys. J., 398, pp. 569–573.
[68] Haensel, P. and M. Proszynski (1982) “Pion condensation in cold dense matter and neutron stars,” Astrophys. J., 258, pp. 306–320.
105 [69] Press, W. (2002) Numerical Recipes in C++: The Art of Scientific Com- puting, Cambridge University Press, New York. URL http://books.google.com/books?id=mSLhDt_XIUQC
[70] Hartle, J. B. (1967) “Slowly Rotating Relativistic Stars. I. Equations of Structure,” Astrophys. J., 150, p. 1005.
[71] Lockitch, K. H. (1999) Stability and rotational mixing of modes in Newto- nian and relativistic stars, Ph.D. thesis, THE UNIVERSITY OF WISCONSIN - MILWAUKEE.
[72] Arfken, G., H. Weber, and F. Harris (2005) Mathematical Methods For Physicists International Student Edition, Elsevier Science, Oxford. URL http://books.google.com/books?id=tNtijk2iBSMC
[73] Kutner, M. (2005) Applied Linear Statistical Models, McGraw-Hill/Irwin series, McGraw-Hill Irwin, New York.
[74] Douchin, F. and P. Haensel (2001) “A unified equation of state of dense matter and neutron star structure,” Astron. and Astrophys., 380, pp. 151–167.
[75] Akmal, A., V. R. Pandharipande, and D. G. Ravenhall (1998) “Equa- tion of state of nucleon matter and neutron star structure,” Phys. Rev. C, 58, pp. 1804–1828. URL http://link.aps.org/doi/10.1103/PhysRevC.58.1804
[76] Wette, K. (2010) Gravitational waves from accreting neutron stars and Cassiopeia A, Ph.D. thesis, Australian National University.
[77] Prix, R. (2007) “Search for continuous gravitational waves: Metric of the multidetector F-statistic,” Phys. Rev. D, 75, p. 023004. URL http://link.aps.org/doi/10.1103/PhysRevD.75.023004
[78] Whitbeck, D. M. Observational Consequences of Gravitational Wave Emis- sion From Spinning Compact Sources, Ph.D. thesis.
[79] Wette, K., B. J. Owen, B. Allen, M. Ashley, J. Betzwieser, N. Christensen, T. D. Creighton, V. Dergachev, I. Gholami, E. Goetz, R. Gustafson, D. Hammer, D. I. Jones, B. Krishnan, M. Landry, B. Machenschalk, D. E. McClelland, G. Mendell, C. J. Messenger, M. A. Papa, P. Patel, M. Pitkin, H. J. Pletsch, R. Prix, K. Riles, L. Sancho de la Jordana, S. M. Scott, A. M. Sintes, M. Trias, J. T. Whelan, and G. Woan (2008) “Searching for gravitational waves from Cassiopeia A with LIGO,” Classical and Quantum Gravity, 25(23), 235011.
106 [80] Prix, R. (2009), “The F-statistic and its implementation in ComputeFStatistic_v2,” LIGO Technical Document T0900149, https://dcc.ligo.org.
[81] Patel, P., X. Siemens, R. Dupuis, and J. Betzwieser (2010) “Implemen- tation of barycentric resampling for continuous wave searches in gravitational wave data,” Phys. Rev. D, 81(8), 084032, 0912.4255.
[82] Lyons, R. (2010) Understanding Digital Signal Processing, Prentice-Hall accounting series, Pearson Education. URL http://books.google.com/books?id=UBU7Y2tpwWUC
[83] Abbott, B. et al. (2004) “Setting upper limits on the strength of periodic gravitational waves using the first science data from the GEO 600 and LIGO detectors,” Phys. Rev. D, 69, p. 082004.
[84] Abbott, B. (2005) “Limits on Gravitational-Wave Emission from Selected Pulsars Using LIGO Data,” Phys. Rev. Lett., 94, p. 181103. URL http://link.aps.org/doi/10.1103/PhysRevLett.94.181103
[85] ——— (2008) “Beating the Spin-Down Limit on Gravitational Wave Emission from the Crab Pulsar,” The Astrophysical Journal Letters, 683(1), p. L45. URL http://stacks.iop.org/1538-4357/683/i=1/a=L45
[86] Abadie, J. et al. (2011) “Beating the Spin-down Limit on Gravitational Wave Emission from the Vela Pulsar,” Astrophys. J., 737, 93, 1104.2712.
[87] Abbott, B. P. et al. (2010) “Searches for gravitational waves from known pulsars with Science Run 5 LIGO data,” Astrophys. J., 713, p. 671.
[88] Aasi, J. et al. (2014) “Gravitational waves from known pulsars: results from the initial detector era,” Astrophys. J., 785, p. 119, 1309.4027.
[89] Abbott, B. e. a. (2005) “First all-sky upper limits from LIGO on the strength of periodic gravitational waves using the Hough transform,” Phys. Rev. D, 72, p. 102004. URL http://link.aps.org/doi/10.1103/PhysRevD.72.102004
[90] Abbott, B. (2007) “Searches for periodic gravitational waves from unknown isolated sources and Scorpius X-1: Results from the second LIGO science run,” Phys. Rev. D, 76, p. 082001. URL http://link.aps.org/doi/10.1103/PhysRevD.76.082001
[91] ——— (2008) “All-sky search for periodic gravitational waves in LIGO S4 data,” Phys. Rev. D, 77, p. 022001. URL http://link.aps.org/doi/10.1103/PhysRevD.77.022001
107 [92] ——— (2009) “Einstein@Home search for periodic gravitational waves in LIGO S4 data,” Phys. Rev. D, 79, p. 022001. URL http://link.aps.org/doi/10.1103/PhysRevD.79.022001
[93] Abbott, B. P. et al. (2009) “All-sky LIGO Search for Periodic Gravita- tional Waves in the Early S5 Data,” Phys. Rev. Lett., 102, p. 111102. [94] ——— (2009) “Einstein@Home search for periodic gravitational waves in early S5 LIGO data,” Phys. Rev. D, 80, p. 042003.
[95] Abadie, J. et al. (2012) “All-sky Search for Periodic Gravitational Waves in the Full S5 LIGO Data,” Phys. Rev. D, 85, p. 022001, 1110.0208.
[96] Abbott, B. et al. (2007) “Upper limits on gravitational wave emission from 78 radio pulsars,” Phys. Rev. D, 76, p. 042001.
[97] Abbott, B. (2007) “Upper limit map of a background of gravitational waves,” Phys. Rev. D, 76, p. 082003. URL http://link.aps.org/doi/10.1103/PhysRevD.76.082003
[98] Abadie, J. et al. (2011) “Directional Limits on Persistent Gravitational Waves Using LIGO S5 Science Data,” Phys. Rev. Lett., 107(26), 271102, 1109.1809. [99] ——— (2010) “First search for gravitational waves from the youngest known neutron star,” Astrophys. J., 722, pp. 1504–1513.
[100] Green, D. A. (2009) “A revised Galactic supernova remnant catalogue,” Bull. Astron. Soc. India, 37, pp. 45–61, 0905.3699. [101] ——— (2014) “A catalogue of 294 Galactic supernova remnants,” ArXiv e-prints, 1409.0637.
[102] Owen, B. J. et al. (2014) in preparation. [103] Owen, B. J. (2010) “How to adapt broad-band gravitational-wave searches for r-modes,” Phys. Rev. D, 82(10), 104002.
[104] Aasi, J. e. a. “SEARCHES FOR CONTINUOUS GRAVITATIONAL WAVES FROM NINE YOUNG SUPERNOVA REMNANTS,” .
[105] Owen, B. J. (1996) “Search templates for gravitational waves from inspiraling binaries: Choice of template spacing,” Phys.Rev., D53, pp. 6749–6761, gr-qc/ 9511032.
[106] Arzoumanian, Z., S. Safi-Harb, T. L. Landecker, R. Kothes, and F. Camilo (2008) “Chandra Confirmation of a Pulsar Wind Nebula in DA 495,” Astrophys. J., 687, pp. 505–515, 0806.3766.
108 [107] Willis, A. G. (1973) “Observations of galactic supernova remnants at 1.7 and 2.7 GHz.” Astronomy and Astrophysics, 26, pp. 237–255.
[108] Kothes, R., T. L. Landecker, and M. Wolleben (2004) “H I Absorption of Polarized Emission: A New Technique for Determining Kinematic Distances to Galactic Supernova Remnants,” Astrophys. J., 607, pp. 855–864.
[109] Kothes, R., T. L. Landecker, W. Reich, S. Safi-Harb, and Z. Ar- zoumanian (2008) “DA 495: An Aging Pulsar Wind Nebula,” Astrophys. J., 687, pp. 516–531, 0807.0811.
[110] Park, S., O. Kargaltsev, G. G. Pavlov, K. Mori, P. O. Slane, J. P. Hughes, D. N. Burrows, and G. P. Garmire (2009) “Nonthermal X-Rays from Supernova Remnant G330.2+1.0 and the Characteristics of its Central Compact Object,” Astrophys. J., 695, pp. 431–441, 0809.4281.
[111] McClure-Griffiths, N. M., A. J. Green, J. M. Dickey, B. M. Gaensler, R. F. Haynes, and M. H. Wieringa (2001) “The Southern Galactic Plane Survey: The Test Region,” Astrophys. J., 551, pp. 394–412, astro-ph/0012302.
[112] Torii, K., H. Uchida, K. Hasuike, H. Tsunemi, Y. Yamaguchi, and S. Shibata (2006) “Discovery of a Featureless X-Ray Spectrum in the Supernova Remnant Shell of G330.2+1.0,” Pub. Astron. Soc. Japan, 58, pp. L11–L14, astro-ph/0601569.
[113] Abadie, J., B. P. Abbott, R. Abbott, M. Abernathy, C. Adams, R. Adhikari, P. Ajith, B. Allen, G. Allen, E. Amador Ceron, and et al. (2010) “First Search for Gravitational Waves from the Youngest Known Neutron Star,” Astrophys. J., 722, pp. 1504–1513.
[114] Johnson-McDaniel, N. K. (2013) “Gravitational wave constraints on the shape of neutron stars,” Phys. Rev. D, 88(4), p. 044016, 1303.3259.
[115] Lindblom, L., B. J. Owen, and S. M. Morsink (1998) “Gravitational radiation instability in hot young neutron stars,” Phys. Rev. Lett., 80, pp. 4843–4846, gr-qc/9803053.
[116] Johnson-McDaniel, N. K. and B. J. Owen (2013) “Maximum elastic deformations of relativistic stars,” Phys. Rev. D, 88, p. 044004, 1208.5227.
[117] Bondarescu, R., S. A. Teukolsky, and I. Wasserman (2009) “Spinning down newborn neutron stars: nonlinear development of the r-mode instability,” Phys. Rev. D, 79, p. 104003, 0809.3448.
109 Vita Ashikuzzaman Idrisy
Ashikuzzaman Idrisy was born in Bangladesh, on August 30, 1987. He attended The City College of New York and obtained a B.S. in Physics and graduated summa cum laude. Ashik entered The Pennsylvania State University in 2009 and joined Benjamin Owen’s research group in 2010 where he studied neutron stars and gravitational waves. While attending The Pennsylvania State University Ashik also lectured for Physics 213/214 during the summer of 2013. A summary of his duties as a researcher and lecturer are presented below. Research Experience
• Lead the development of an automated gravitational wave search analysis pipeline to improve and optimize end-user experience • Mined terabyte sized data sets on high performance supercomputing clusters to find gravitational wave signals buried in the noise • Conducted R&D on resampling algorithm to reduce the computational cost of gravitational wave analysis pipeline • Calculated r-mode frequencies for stellar models using realistic equations of state • Numerical solved differential equations with techniques such as finite-difference and spectral method
Teaching Experience
• Managed and organized teaching assistants in creatively engaging students via exam and homework preparation • Collaborated with departmental administrators to produce course content such as lecture presentations, and laboratory activities • Increased student comprehension by writing companion to course textbook • Experimented with handwritten exams to refine students’ abilities to commu- nicate their knowledge of the course content