GRAVITATIONAL SIGNATURE of CORE-COLLAPSE SUPERNOVA RESULTS of CHIMERA SIMULATIONS by Konstantin Yakunin

GRAVITATIONAL SIGNATURE OF CORE-COLLAPSE SUPERNOVA RESULTS OF CHIMERA SIMULATIONS by Konstantin Yakunin

A Dissertation Submitted to the Faculty of The Charles E. Schmidt College of Science in Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy

Florida Atlantic University Boca Raton, FL August 2011 Copyright by Konstantin Yakunin 2011

ii GRAVITATIONAL SIGNATURE OF CORE-COLLAPSE SUPERNOVA RESULTS OF CHIMERA SIMULATIONS by Konstantin Yakunin

This dissertation was prepared under the direction of the candidate’s dissertation advisor, Dr. Pedro Marronetti, Department of Physics, and has been approved by the members of his supervisory committee. It was submitted to the faculty of the Charles E. Schmidt College of Science and was accepted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy.

SUPERVISORY COMMITTEE:

Pedro Marronetti, Ph.D. Dissertation Advisor

Stephen W. Bruenn, Ph.D. Dissertation Co-Advisor

Warner A. Miller, Ph.D.

William Kalies, Ph.D. Warner A. Miller, Ph.D. Chair, Department of Physics

Gary W. Perry, Ph.D. Dean, The Charles E. Schmidt College of Science

Barry T. Rosson, Ph.D. Date Dean, Graduate College

iii ACKNOWLEDGEMENTS

I am deeply thankful to my thesis advisor, Dr. Pedro Marronetti who guided my studies during the PhD program. My achievements wouldn’t have been possible without his support and understanding. His sage advices, insightful criticisms, and patient encouragement aided the writing of this thesis in innumerable ways. I would also express my gratitude to my co-advisor Dr. Steven W. Bruenn whose steadfast support of this project was greatly needed and deeply appreciated. I deeply thank Dr. Shin Yoshida for the smooth and enjoyable collaboration on the supernova code and gravitational wave code. I am grateful to Dr. Antony Mezzacappa and all present and past members of CHIMERA collaborations for giving me the possibility to do this thesis. I am grateful to my committee members Dr. Warner Miller, Dr. William Kalies for providing me with valuable feedback, and I greatly appreciate ﬁnancial support for this work by National Science Foundation (grant NSF-PHYS-0855315). I thank my teachers, Elena Evgenievna Kazakova, Dr. Valery Petrovich Beskachko, Dr. Christopher Beetle, Dr. Shen Li Qiu, and Dr. Sam Faulkner for sharing their knowledge with me. I thank my friend Cyndee Finkel for her constant altruistic support and for proof-reading. Lastly, I oﬀer my regards and blessings to everybody at Department of Physics who supported me in any respect during these years.

iv ABSTRACT

Author: Konstantin Yakunin Title: Gravitational Signature of Core-Collapse Supernova Results of CHIMERA Simulations Institution: Florida Atlantic University Dissertation Advisor: Dr. Pedro Marronetti Degree: Doctor of Philosophy Year: 2011

Core-collapse supernovae (CCSN) are among the most energetic explosions in the universe, liberating ∼ 1053 erg of gravitational binding energy of the stellar core. Most of this energy (∼ 99%) is emitted in neutrinos and only 1% is released as electromagnetic radiation in the visible spectrum. Energy radiated in the form of gravitational waves (GWs) is about ﬁve orders smaller. Nevertheless, this energy corresponds to a very strong GW signal and, because of this CCSN are considered as one of the prime sources of gravitational waves for interferometric detectors. Gravita- tional waves can give us access to the electromagnetically hidden compact inner core of supernovae. They will provide valuable information about the angular momentum distribution and the baryonic equation of state, both of which are uncertain. Fur- thermore, they might even help to constrain theoretically predicted SN mechanisms. Detection of GW signals and analysis of the observations will require realistic signal predictions from the non-parameterized relativistic numerical simulations of CCSN. This dissertation presents the gravitational wave signature of core-collapse

v supernovae. Previous studies have considered either parametric models or non- exploding models of CCSN. This work presents complete waveforms, through the explosion phase, based on first-principles models for the first time. We performed 2D simulations of CCSN using the CHIMERA code for 12, 15, and 25M non-rotating progenitors. CHIMERA incorporates most of the criteria for realistic core-collapse modeling, such as multi-frequency neutrino transport coupled with relativistic hydrodynamics, effective GR potential, nuclear reaction network, and an industry-standard equation of state. Based on the results of our simulations, I produced the most realistic gravitational waveforms including all postbounce phases of core-collapse supernovae: the prompt convection, the stationary accretion shock instability, and the corresponding explosion. Additionally, the tracer particles applied in the analysis of the GW signal reveal the origin of low-frequency component in the prompt part of gravitational waveform. Analysis of detectability of the GW signature from a Galactic event shows that the signal is within the band-pass of current and future GW observatories such as AdvLIGO, advanced Virgo, and LCGT.

vi DEDICATION

This thesis is dedicated to my parents, Nikolay Alekseevich Yakunin and Liudmila Vladimirovna Yakunina CONTENTS

List of Figures ...... x

1 Introduction ...... 1

2 Supernovae ...... 4 2.1 Classiﬁcation of Supernovae...... 4 2.2 Core-Collapse supernovae rates...... 7 2.3 The mechanism of core collapse supernova...... 9

3 Gravitational Waves ...... 20 3.1 General Relativity...... 21 3.2 Linearized Theory...... 21 3.3 Wave solution in vacuum...... 22 3.4 Polarization of Gravitational Waves...... 23 3.5 Gravitational Radiation in the Weak-Field Slow-Motion Limit.... 25 3.6 Gravitational Wave Astronomy...... 25 3.6.1 Estimation of the detectability of gravitational signals from CCSN 25 3.6.2 Gravitational waves detectors...... 26

4 CHIMERA Code ...... 33 4.1 Hydrodynamics in CHIMERA...... 34 4.2 Gravity in CHIMERA...... 38 4.2.1 Derivation of eﬀective GR potential...... 40

vii 4.2.2 Implementation...... 41 4.2.3 Tests of the eﬀective GR potential...... 42 4.3 Neutrino Transport...... 45 4.4 Nuclear Reaction Network...... 46 4.5 Equations of State...... 49 4.6 Tracer particle method...... 51

5 Gravitational Wave Analysis ...... 53 5.1 Mass quadrupole of gravitational wave...... 53 5.2 Gravitational waves produced by anisotropic neutrino emission.... 55 5.2.1 Deﬁnition of angles...... 57 5.3 Direction dependent neutrino luminosity dL/dΩ...... 60 5.4 Numerical implementation...... 61

5.4.1 Computation of N20 ...... 61 5.5 Veriﬁcation tests...... 61 5.5.1 Oscillating ring around a central mass...... 61 5.5.2 Rotating triaxial ellipsoid...... 64

6 Results ...... 66 6.1 Gravitational Wave Emission by Aspherical Mass Motions...... 66 6.1.1 Prompt signal...... 68 6.1.2 Quiescent stage...... 72 6.1.3 Strong signal...... 72 6.1.4 The “tail”part...... 74 6.2 Gravitational Wave Emission by Anisotropic Neutrino Radiation... 84

7 Conclusions and Future Work ...... 90

viii A Tensor spherical harmonics flm ...... 95

A.1 Explicit form of Wlm and Xlm ...... 95 A.2 Tensor spherical harmonics...... 96

B Transverse traceless part of “directional”tensor ...... 97

C Analytical tests of the GW code...... 99 C.1 Rotating ellipsoid test...... 99 C.2 Oscillating Ring Test...... 104

Bibliography ...... 107

ix LIST OF FIGURES

2.1 The current classification scheme of supernovae. Type Ia SNe are associated with the thermonuclear explosion of accreting white dwarfs. Other SN types are associated with the core collapse of massive stars. Some type Ib/c and IIn SNe with explosion energies E > 1052 erg are often called hypernovae...... 5 2.2 The spectra of the main SN types at maximum, three weeks, and one year after maximum. The representative spectra are those of SN1996X for type Ia, of SN1994I (left and center) and SN1997B (right) for type Ic, of SN1999dn (left and center) and SN1990I (right) for type Ib, and of SN1987A for type II. At late times (especially in the case of the type Ic SN1997B) the contamination from the host galaxy is evident as an underlying continuum plus unresolved emission lines. In all figures of this paper the spectra have been transformed to the parent galaxy rest frame...... 6 2.3 The distribution of number of the detected CCSNe events with the distance during the initial LIGO epoch (2002 - 2010)...... 9 2.4 Remnants of massive single stars as a function of initial metalicity and initial mass. In the regions above the thick green line (for the higher initial metalicity), the hydrogen envelope is stripped during its evolution due to the active mass loss processes. The dashed blue line indicates the border of the regime of direct black hole formation. The white strip near the right lower corner indicates the occurrence of the pair-instability supernovae. In the white region on the left side at lower mass, the stellar cores do not collapse and end their lives as white dwarfs. This figure is taken from Heger et al (2003) [1]..... 10 2.5 The onion-skin structure of a massive, evolved star just prior to core collapse. (Not to scale)...... 12

x 2.6 Schematic representation of the evolutionary stages from stellar core collapse through the onset of the supernova explosion to the neutrino- driven wind during the neutrino-cooling phase of the PNS. Mhc means the mass of the subsonically collapsing, homologous inner core. RFe , Rs , Rg , Rns , and Rν being the iron core radius, shock radius, gain radius, neutron star radius, and neutrinosphere, respectively. This figure is taken from [2]...... 15 2.7 Schematic representation of the processes in supernova core during the shock revival phase. This figure is taken from [3]...... 16 2.8 Images of the gas entropy (red is higher than the equilibrium value, blue is lower) illustrate the instability of a spherical standing accretion shock. This model has γ = 4/3 and is perturbed by placing overdense rings into the infalling preshock gas. The shock is kept stalled by using a cooling function. Note that with the scaling for a realistic supernova model, the last image on the right corresponds to 300 ms. These simulations are axisymmetric, forcing a reflection symmetry about the vertical axis. This figure is taken from [4]...... 19

3.1 Gravitational wave , which propagates along the +z axis. (a) + polarization; (b) × polarization. Figure taken from [5]...... 24 3.2 Simpliﬁed scheme of a light-interferometric gravitational wave detector. Figure taken from [6]...... 27 3.3 Model curves of optimal sensitivity of Advanced LIGO for diﬀerent type of GW sources. The models take into considerations the principal contributors of the noise. Figure taken from [6]...... 31

4.1 Convergence factor as obtained from the convergence study with the diﬀerent angular resolutions h (64 angular zones), 2h (32 angular zones), 4h (16 angular zones) and ﬁxed radial resolution (128 zones). Conver- gence factor of 3 corresponds to 1.6 order algorithm. The possible reason of the lower convergence is a presence of a sharp discontinuity at the shock surface. One can expect to see the second order convergence higher resolutions in both space directions...... 37

xi 4.2 Time evolution of the central density ρc for Model AB-GR ( black solid line), Marek et al. (black dotted line), CHIMERA eff. potential (Coordinate time) (green solid line), CHIMERA eff. potential (Proper time) (green dashed line), CHIMERA Newtonian potential (Coordinate time) (pink solid line), The left panel shows the collapse phase, while the right panel shows the post-bounce evolution. Note the different axis scales in both panels...... 42 4.3 Comparison of the radial velocity profile at 100 ms between CHIMERA simulation with the effective GR potential and 1D full GR code AGILE- BolzTran simulations...... 43 4.4 Normalized eigenfrequency of fundamental radial mode of non-rotating relativistic stars (n = 1 polytrope) plotted as a function of compactness parameter M/R: ”original effective“ means original form of the effective potential, ”Γ = 1 in Equil“ means applying Lorentz factor Γ only in computing equilibrium. In ”Marek et al. Case (A)“, Lorentz factor is introduced both in equilibrium and in perturbation...... 44 4.5 Schematic of CHIMERA’s assumed geometry. The hydrodynamics is evolved in the three-dimensional space defined by r, θ, φ. At each point along each r, a neutrino phase space is evolved as well, spanned by 2 neutrino propagation angles, Θ and Φ, and a neutrino energy, . The heart of the ray-by-ray-plus approximation lies in the assumption that each ray, r, can, for the most part, be decoupled from neighboring rays during the transport step, requiring only integral corrections (e.g. lateral pressure gradients). We assume the imposed boundary conditions on each radial transport solution mimic the effects of neighboring rays. The additional degrees of freedom provided by the nuclear network are not shown. Diagram from [7]...... 47 4.6 A portion of the nuclear N-Z plane important for supernova nucleosynthesis. The color map gives the abundance of each species at the indicated density and temperature (roughly 6 × 105 g/cm3 and 4.7 billion K, respectively). The locations of the current 14 isotope network species are marked with αs. The extents of the 150 isotope network are circumscribed by the dashed line, exhibiting the good, but not complete, coverage for the chosen conditions. Figure taken from [7].. 50

4.7 The distribution of tracer particles for the 12M model: left panel: at t = 235 ms, center panal: at t = 240 ms, right panel: at t = 440 ms elapsed time. Figure from [8]...... 51

5.1 Schematic ﬁgure of coordinate frames...... 56

xii 5.2 Angular dependence of Ψ in Eq. (5.10). Note that the angle is measured from the symmetry axis. Figure is taken from [9]...... 57 5.3 Oscillating ring test...... 62

5.4 A20 amplitude of gravitational wave caused by ring oscillations. The central mass is M = 10M , the ring mass is m = 1M , the equilibrium radius of ring is R0 = 6.0Rg, and the thickness of the ring is 0.1R0. The amplitude is α = 0.2. The grid points are nx = 512 in radial direction, and ny = 512 in angular direction (0 ≤ ϑ0 ≤ π)...... 63 5.5 Ellipsoid rotating around z-axis...... 64

5.6 rh+ and rh× signal from rotating triaxial ellipsoid. The central density is 1014g/cm3, three axes are a = 3 × 106(cm), b = 2 × 106(cm), c = 106(cm). Rotational frequency is Ω = 0.5(/sec). The observer’s position angle is ϑ = π/6...... 65

6.1 The left column shows the GW strain times the distance to the observer vs. post-bounce time for non-rotating progenitors of 12, 15, and 25 M . The signal is split into matter- (red-solid) and neutrino-generated (blue-dashed) signals. Note that the scales are diﬀerent for these two signals. The insets show the ﬁrst 70 ms after bounce. The right column shows the corresponding characteristic strain for both the matter (red) and the total (black) signals, compared to the AdvLIGO sensitivity curve. Figure from [10]...... 69

6.2 Top: Gravitational wave strain h+ times the distance to the observer r vs. post-bounce time for the 15M non-rotating progenitor model. Below, entropy distribution snapshots typical of the prompt, strong, and tail stages of the signal. Note the difference in scale of the left snapshot and two others. Figure taken from [11]...... 70 6.3 Contributions to the GW signal from two different regions for the 15 M model: the PNS (r < 30 km) and the region above the PNS (r > 30km). The latter includes the region of neutrino-driven convection, the SASI, and the shock...... 71 6.4 Left: Trajectories of the tracer particles. It is shown the clear deflec- tions of infalling particles through the shock that collectively produce low-frequency high- amplitude component of the GW signal shown on the right panel.Right: Comparison between the matter signal (solid red) and signal calculated using the tracers (dashed blue). Both panels correspond to our 15 M simulation...... 72

xiii 6.5 Top: Entropy distribution at 244 ms after bounce for the 15 M model. A large, low-entropy (blue-green) accretion funnel at an angle quasi- orthogonal to the symmetry axis and high-entropy (yellow-orange-red) outﬂows below the shock, along the symmetry axis, are evident. Bot- tom: Shock radius as a function of time for three regions: the north pole (solid blue), the equatorial plane (dotted black), and the south pole (dashed red)...... 73

6.6 Left: Trajectories of the tracer particles for 15 M simulation. The trajectories show how initially accreting matter is ejected by the revived shock and produces “tail”part of the GW signal...... 74 6.7 Snapshots of the entropy distribution during explosion for the three models representing each type of the explosion. The GW signal may indicate that the explosion is “spherical”, “oblate”, or “spherical”. Fig- ure taken from [12]...... 75

6.8 Gravitational wave quadrupole amplitudes A20 as functions of post- bounce time (top) and corresponding Fourier spectra (bottom) associated with mass motions for 15M model. Simulations are performed by Marek et al. [13]. Figure taken from [13]...... 77 6.9 Luminosities of electron neutrinos versus time as measurable for a distant observer located along the polar axis of the 2D spherical coordinate grid. Comparison between 15M simulation of Garching group [13](top) and CHIMERA simulation (bottom). The top panel of Figure taken from [13]...... 79

6.10 A sample of GW strain (h+ ) times the distance, D, vs. time after bounce. This signal was extracted from a simulation of Murphy et al. using a 15 M progenitor model (Woosley & Heger 2007) and an 52 electron-type neutrino luminosity of Lνe = 3.7 × 10 erg/s. Figure taken from [12]...... 80

6.11 Trajectories of the tracer particles for 15 M simulation. The color lines track trajectories of the particles that correspond to the downﬂow plume hitting the PNS surface and bouncing back...... 81

6.12 hchar (Eq. 3.24) vs. frequency for the suite of simulations presented in Murphy et al. The spectra show broad peaks and some dependence upon the progenitor mass: ∼300 Hz for 12 M and ∼400 Hz for 40 M . For comparison, the approximate noise thresholds for Initial LIGO (solid-black curve), enhanced LIGO (dot-dashed-black curve) and Advanced LIGO (dashed-black curve) are plotted. Figure taken from [12]...... 83

xiv 6.13 Energy emitted by GWs during the ﬁrst 500 ms after bounce for all three models presented here...... 84

6.14 Snapshots of the total neutrino luminosity for the 15 M model. At the top panel you can see almost spherical symmetric distribution of the neutrino luminosity at 125 ms after bounce. The middle panel shows the clear asymmetry in the neutrino luminosity due to the accretion downﬂow that increases neutrino opacities at the equatorial zone at 381 ms after bounce. The bottom panel illustrates the distribution of the neutrino luminosity at 533 after bounce that leads to decreasing in the amplitude of the GW signal...... 85

E2 6.15 Gravitational wave quadrupole amplitude A20 by Müllleret al. [14] vs. time (post-bounce) due to convective mass flow and anisotropic neutrino emission (thin line) for the rotating delayed explosion model s15r of Buras et al. (2003) [15]. The inset shows an enlargement of the signal around the time of bounce. Figure taken from [14]...... 87 6.16 Gravitational wave signals are produced in the simulations of Kotake et al. [16] The top panel shows the GW spectrum contributed from neutrinos (solid) and from the matter (dashed) in a rotating model −1 with Ω = 4 rad imposed initially on a 15M progenitor model. In the bottom panel, the open circles and the pluses represent the amplitudes of hν, eq with the characteristic frequencies of νeq for the models with the cylindrical and the shell-type rotation profiles, respectively. Under the frequency of νeq, the GWs from the neutrinos dominate over those from the matter contributions. From the panel, it is seen that the GWs from neutrinos dominate over the ones from the matter in a lower frequency (f ≤ 100 Hz). Note that the source is assumed to be located at the distance of 10 kpc. Figure from [16]...... 88 6.17 Gravitational waveforms produced by the 3D simulations of Kotake et al. [17] from neutrinos (bottom) and from the sum of neutrinos and matter motions (top), seen from the polar axis and along the equator (indicated by “Pole”and “Equator”) with polarization (+ or × modes). The distance to the SN is assumed to be 10 kpc. Figure from [17]... 89

xv CHAPTER 1

INTRODUCTION

Supernovae are most powerful explosions in nature. The total gravitational binding energy released by a core-collapse supernova is typically 1053 erg. 99% percent of the total supernova energy is emitted in the form of neutrinos, which are barely detectable, unless the core-collapse supernova occurs in the Local Group. About 1%, ∼ 1051 erg, goes into the kinetic and internal energy of the supernova ejecta, a fraction of which is converted into electromagnetic radiation [18]. The general picture of stellar evolution and of the nuclear fusion history of massive stars is relatively well established. We understand that the electron-degenerate iron core of massive star become unstable to electron capture and endothermic photodissociation of heavy nuclei. Collapse starts and continues once the inner core has reached nuclear densities ∼ 1014 g/cm3. At this point, nuclear repulsive forces lead to a stiﬀening of the nuclear equation of state. As a result, the collapse halts and a shock wave forms. The outgoing shock wave quickly loses its energy through the dissociation of material in the outer core and due to neutrino emission. The shock stalls. Up to this point, the a supernova mechanism is understood. But a question remains: how is the stalled shock revived to ﬁnally explode the star? It is obvious that a part of the gravitational binding energy contained in the newborn hot protoneutron star must be converted into the kinetic and internal energy of the exploding mantle. The detailed mechanism of core-collapse supernova explosions is one of the out-

1 standing problems in astrophysics and it has resisted more than forty years of specu- lation, concerted theoretical work, and numerical exploration [2,18–20] and references therein. Most of what we know about core-collapse supernovae is based on observations in the electromagnetic spectrum [21]. For the ﬁrst time, neutrino emission from a supernova was directly measured from Earth on 24 February 1987 when neutrino detectors recorded about 20 neutrinos from SN1987A in the LMC [21]. Modern day detectors are projected to detect thousands of neutrinos from a supernova in the Local Group [22]. However, besides via electromagnetic waves and neutrinos there is a third way in which physical information from a core-collapse supernova can reach observers on Earth: Gravitational waves, predicted by Einsteins theory of gravity, General Rela- tivity (GR) [23]. Gravitational waves are propagating vibratory disturbances, ripples, in space-time, traveling through the universe at the speed of light. They are generated by time-changing mass multipole moments related to coherent, bulk motions of huge amounts of mass-energy at frequencies related to the dynamical timescale of their emitters. In contrast to electromagnetic waves, gravitational waves cannot be used to form an image of the radiating system since their wavelengths are comparable to or larger than their coherent, bulk-moving sources. Instead, gravitational waves are akin to sound and carry a stereophonic description of their sources dynamics in two independent polarizations. Gravitational waves interact very weakly with matter, thus can travel nearly unscathed from their source through intervening matter to distant observers. This makes them ideal carriers of information, but also means that gravitational waves interact very weakly with detectors on Earth. To date, gravitational waves have never been observed directly, but there is strong evidence supporting their existence from the observed orbital shrinking of the

2 Hulse-Taylor binary system [24]. The international array of second-generation large- scale light-interferometric gravitational wave observatories (LIGOs) is in development stages. Hopefully they will reach fully operational state and design sensitivities in 2015. Current estimations [22] suggest that a galactic supernova would likely be visible to the Advanced LIGO. The physical information carried by gravitational waves emitted in a core-collapse supernova could be an important, possibly crucial piece in the supernova puzzle and may also help to form a complete picture of the core-collapse supernova. However, detailed and accurate theoretical predictions of the gravitational wave signature of core-collapse supernovae and a good theoretical understanding of the various possible gravitational wave emission processes in a core-collapse supernova will be essential for the extraction of physical information from an observed supernova gravitational wave signal [22,25].

3 CHAPTER 2

SUPERNOVAE

2.1 CLASSIFICATION OF SUPERNOVAE

The classification of SNe is generally performed on the optical spectra and also on their light curves. The SN types are assigned on the basis of the chemical and physical properties of the outermost layers of the exploding stars. The classification scheme and the general spectral characteristics of the main types of SNe are presented on Figure 2.1 and Figure 2.2. (Both figures are taken from [26]).

Type Ia Type Ia SNe play a crucial role in Cosmology with their high luminosity and relatively small luminosity dispersion at maximum. SNIa are discovered in all types of galaxies and are not associated with the arms of spirals as strongly as other SN types. The spectra are characterized by lines of intermediate mass elements such as calcium, oxygen, silicon and sulfur during the peak phase and by the absence of H at any time (see Figure 2.2). With age the contribution of the Fe lines increases, and several months past maximum, the spectra are dominated by [Fe II] and [Fe III] lines [26]. SNIa are associated with the thermonuclear explosion of a white dwarf. The peak luminosity of SNIa directly depends on the amount of radioactive 56Ni produced in the explosion [27]. A correlation between the peak luminosity and the shape of the light curve of SNIa is used in determining cosmological distances.

4 Figure 2.1: The current classiﬁcation scheme of supernovae. Type Ia SNe are associated with the thermonuclear explosion of accreting white dwarfs. Other SN types are associated with the core collapse of massive stars. Some type Ib/c and IIn SNe with explosion energies E > 1052 erg are often called hypernovae.

Type Ib and Ic Type Ib and Ic appear only in spiral type galaxies. The SNIb/c exhibit relatively strong radio emission with steep spectral indices, which is produced by the shock interaction with a dense circumstellar medium [28]. These types of SNe are associated with the core collapse of massive stars which have been stripped of their outer H (and possibly He) envelope. Their characterizing features are the absence of H and Si II lines and the presence of He I. The He lines are produced by fast electrons accelerated by γ-rays from the decay of 56Ni and 56Co. Objects that do not show strong He line form the class of helium poor type Ic [26].

Type II Type II SNe are characterized by the clear presence of H in their spectra. They are strongly associated with regions of recent star formation and with the core collapse of massive stars. SNII show a wide variety of properties both in their light curves and in their spectra [29]. Four subclasses of SNII are commonly mentioned in

5 Figure 2.2: The spectra of the main SN types at maximum, three weeks, and one year after maximum. The representative spectra are those of SN1996X for type Ia, of SN1994I (left and center) and SN1997B (right) for type Ic, of SN1999dn (left and center) and SN1990I (right) for type Ib, and of SN1987A for type II. At late times (especially in the case of the type Ic SN1997B) the contamination from the host galaxy is evident as an underlying continuum plus unresolved emission lines. In all ﬁgures of this paper the spectra have been transformed to the parent galaxy rest frame.

6 the literature: IIP, IIL, and IIn in addition to the above mentioned IIb. However, some objects do not fit into any of these classes. SNIIP (Plateau) and SNIIL (Linear) are often referred as normal SNII. The subclassification is made according to the shape of the optical light curves. The luminosity of SNIIP stops declining shortly after maximum. It forms a plateau 2–3 months long during which a shock wave moves through the massive hydrogen envelope [26]. SNIIL show a linear, uninterrupted luminosity decline because of a lower mass envelope. The two classes are not clearly separated. There are a number of intermediate cases with a short plateau. A quantitative criterion for the classification of the light curves has been proposed on the basis of the average decline rate of the first 100 days. Starting 150 days past maximum the luminosity of both types settles into an exponential decline, consistent with complete (or constant) trapping of the energy release of the radioactive decay of 56Co into 56Fe.

2.2 CORE-COLLAPSE SUPERNOVAE RATES

In the last millennium, six supernovae are known in our galaxy and one in the LMC [30] (see table below) As it is estimated, the six recorded supernovae, are only 20% of the galactic supernovae that exploded since 1000 AD. Most of them were either screened from view by the galaxy dust, or could be observed from the southern hemisphere only when there was no record of astronomical events in that part of the Earth. For example, the supernova remnant RX J0852–4642, which is nearby (∼ 0.2 kpc) and exploded at ∼ 1300 AD, was unrecorded [31] and Cas A, a supernova remnant of a 17th-century explosion, was not recognized as a supernova. Estimations of supernova rates can either be made based on galactic observations,

7 Table 2.1: Local Group Supernova Record

Name Year Distance (kpc) Type

SN 1006 1006 2.2 Ia Crab 1054 2.0 II SN 1181 1181 2.6 II Tycho 1575 2.4 Ia Kepler 1604 4.2 Ib/II Cas A 1680 2.92 Ib SN1987A 1987 50 II star formation rates and the initial mass function, or on extragalactic supernova observations, statistics, galaxy morphology and mass [26]. Table 2.2 summarizes the rate estimates.

Table 2.2: SN frequencies in the local group galaxies [32].

Galaxy Distance (kpc) CCSN Rate (100 year)−1

Milky Way < 15 0.50 - 2.50 LMC 50 0.10 0.50 SMC 60 0.06 0.12 M33 770 0.20 1.20 M31 840 0.16 0.68

Total < 103 1.05 - 5.05

Based on Table 2.2 the most optimistic estimation for the CCSN in the Local Group is one event within 20 years. The nearest cluster of galaxies to the Local Group is the Virgo Cluster at a distance of 1020 Mpc. The Virgo Cluster contains a signiﬁcant number of galaxies with high star formation rates. Arnaud et al. [33]

8 Figure 2.3: The distribution of number of the detected CCSNe events with the distance during the initial LIGO epoch (2002 - 2010). estimated a rate of ∼ 5 core-collapse supernovae per year for the Virgo Cluster. That is in a good agreement with the optically detected CCSN events during the initial LIGO epoch (Figure 2.3).

2.3 THE MECHANISM OF CORE COLLAPSE SUPERNOVA

The fate of single massive stars (> 9M ) is determined by its initial mass and by the history of its mass loss during its evolution. The mass loss is affected by the initial metalicity of the star, because the mass loss rate is sensitive to the photon opacity, which is defined by the metalicity. The stars with high initial metalicity have more mass loss, and therefore, have smaller helium cores and hydrogen envelopes during its evolution. Stellar collapse of such stars tends to lead to the formation of a neutron star, while for the lower metalicity stars, a black hole. Figure 2.4 shows how the remnant of massive stars depends on the initial mass and the metalicity (this figure is taken from [1]). From the figure, stellar collapse of the stars with the initial masses above ∼ 9M and below ∼ 25M lead to the formation of neutron stars.

Above 25M , black holes will be formed either by fall-back of matter after the weak explosion (below 40M ) or directly if the stellar core is too massive to produce the

9 Figure 2.4: Remnants of massive single stars as a function of initial metalicity and initial mass. In the regions above the thick green line (for the higher initial metalicity), the hydrogen envelope is stripped during its evolution due to the active mass loss processes. The dashed blue line indicates the border of the regime of direct black hole formation. The white strip near the right lower corner indicates the occurrence of the pair-instability supernovae. In the white region on the left side at lower mass, the stellar cores do not collapse and end their lives as white dwarfs. This ﬁgure is taken from Heger et al (2003) [1].

10 outgoing shock wave (above 40M )[20]. It needs to be emphasized that the statement about the masses of stars that form neutron stars and black holes is speculative, and based on many assumptions. There is no direct evidence for this and other scenarios have been suggested. In this work, we concentrate on the ordinary core-collapse supernova which lead to the neutron star formation (∼ 9M ≤ M ≤ 25M with the solar metalicity). Such stars undergo a sequence of thermonuclear cycles during ∼ 107 yr of their evolution. The process starts with the fusion of hydrogen into helium at a temperature above 2 × 107 K. When hydrogen is exhausted, the core of the star begins to collapse, causing a rise in temperature and pressure, which becomes great enough to ignite the helium and start a helium-to-carbon fusion cycle. The process, repeats several times [4He → 12C → 20Ne → 16O → 28Si (temperature of silicon ignition > 3 × 109K)]. Each time the core collapses, and the collapse is halted by the ignition of a further process involving more massive nuclei and higher temperatures and pressures. The process terminates with the formation of 56Fe, the nucleus with the maximum binding energy per nucleon. As a result, the star has an onion-like structure of shells comprised of elements of progressively lower atomic weight at progressively lower densities and temperatures (Figure 2.5).

During the evolution a central core of mass ∼1.5M , consisting primarily of iron group nuclei, is formed. The size of the iron core is of the order of 109 cm while the stellar radius is larger than 1013 cm. At the core and the surrounding shell, the density decreases steeply. So, the dynamical timescale of the core (τ ∝ (Gρ¯)1/2 with G andρ ¯ being the gravitational constant and the average density) is much shorter than that of the envelope. That is, the dynamics of the iron core is not aﬀected by the envelope [20]. This is why CCSN models consider a dynamics of the core only.

Γ Γ The core is supported primarily by electron degeneracy pressure P ∝ Ye ρ , with

11 Figure 2.5: The onion-skin structure of a massive, evolved star just prior to core collapse. (Not to scale).

Ye = ne/n being the number of electrons per baryon and Γ = 4/3 in the extreme relativistic degeneracy limit [34]. Despite its high central temperature, the core of the massive star has low entropy (∼ 0.52 per barion) because of its high central density (∼ 5 × 109 g/cm3). The contribution of the degenerate electrons to the total

iron core pressure amounts to ∼ 90% in cores of masses in the range of ∼ 1220M [21]. The maximum mass of a nonrotating ﬂuid body that can be held in hydrostatic equilibrium by electron-degeneracy pressure in Newtonian gravity is the Chan- drasekhar mass

2 MCh = 1.457(2Ye) M . (2.1)

The presupernova core of 15M has a core of about 1.5M with initial central temperature Tc,i, central density ρc,i central lepton fraction Yc,i and entropy per baryon si approximately given by [34]:

12  T ≈ 8 × 109K = 0.69MeV/k  c,i   9 3  ρc,i ≈ 3.7 × 10 g/cm  = Mcore ≈ 1.5M ≥ MCh Y ≈ 0.42  c,i    si/k ≈ 0.91  At typical core densities and temperatures, the electron capture on heavy nuclei and free protons [34] is

− e + (Z,A) → νe + (Z − 1,A), (2.2)

− e + p → νe + n. (2.3)

This process decreases the electron pressure support and hence Ye, and the core begins to collapse. Furthermore, the endothermic photodissociation of iron nuclei

γ +56 Fe → 13α + 4n − 124.4MeV (2.4)

leads to the reduction of the thermal pressure support. In addition, the internal energy that is absorbed by the dissociation of nuclei is channeled away from increasing the kinetic energy of the particles and therefore results in a smaller increase in pressure. Both of them promote the core collapse [20]. The electron-type neutrinos created in the captures are able to stream freely out of the collapsing core until densities of about 2 × 1012 g/cm3 are reached. At this point the neutrino scattering opacities become so large that the diﬀusion timescale of the neutrinos becomes longer compared to the dynamical timescale of collapse. This means that the neutrinos can not escape freely from the core and are trapped in the core. The equilibrium of weak interactions

13 − e + p ↔ n + νe ( β-equilibrium) is established, keeping the lepton number Yl =

Ye +Yν ﬁxed [21,34,35]. After this, the entropy is conserved and the collapse proceeds adiabatically. The collapsing core consists of two parts: the (homologically collapsing) inner core and the (supersonically infalling) outer core. Matter inside the sonic point, where the local speed of sound equals the radial collapse velocity, collapses homologously (v ∝ r

self-similar). The inner core encloses ∼ 0.5 − 0.8M of material [36]. On the other hand, the material outside the sonic point (the outer core) falls with v ∝ r−1/2 (Figure 2.6).

14 3 When nuclear densities are reached in the collapsing core (ρc ∼ 3 × 10 g/cm ), the homologous core decelerates and rebounces due to the high incompressibility of the nuclear matter. This drives a shock wave into the outer core, which is still collapsing with velocities ∼ 0.1c. The shock propagating into the outer core dissociates nuclei

− into free nucleons, the electron capture process e + p → n + νe generates a huge amount of electron neutrinos just behind the shock. These neutrinos are trapped until the shock arrives at the edge of neutrino diﬀusion region, so-called neutrinosphere. As the shock wave passes through the neutrinosphere, the previously trapped electron neutrinos decouple from the matter and begin to free-stream. This sudden release of electron neutrinos forms a prompt (duration ∼ 20 ms), ultra-luminous (∼ 1054 erg/s ) burst of neutrinos, which is called the neutronization burst (Figure 2.6). The total energy emitted in the neutronization burst is only of the order of 1051 erg due to the short duration timescale. This electron-neutrino breakout signal is expected to be detected from the Galactic supernova in modern neutrino detectors such as SuperKamiokande and the IceCube Neutrino Observatory [20]. The bounce shock stalls within only 10 − 20 ms of its birth and turns into quasi- stationary accretion shock at radii of 100 − 200 km as the result of the following

14 Figure 2.6: Schematic representation of the evolutionary stages from stellar core collapse through the onset of the supernova explosion to the neutrino-driven wind during the neutrino-cooling phase of the PNS. Mhc means the mass of the subsonically collapsing, homologous inner core. RFe , Rs , Rg , Rns , and Rν being the iron core radius, shock radius, gain radius, neutron star radius, and neutrinosphere, respectively. This ﬁgure is taken from [2].

15 Figure 2.7: Schematic representation of the processes in supernova core during the shock revival phase. This ﬁgure is taken from [3] two eﬀects. First, as the shock propagates outwards, it loses its strength to the dissociation of outer-core material into nucleons at a cost of ∼8.8 MeV per nucleon [21]. Second, and most importantly, as the shock dissociates nuclei into free nucleons, it loses its energy due to emission of a large amount of the electron neutrinos via

− e + p → n + νe [34] (Figure 2.6). Within a short time (∼ 50 ms) a thermodynamic profile is established (Figure 2.7). A hot and dense proto-neutron star (PNS) has formed. It slowly contracts while deleptonizing and cooling as neutrinos of all flavors diffuse out on timescale of several hundred ms. Convective processes may increase neutrino cooling and therefore neutrino luminosity [37]. If the energy transfer from the neutrinos to the material near the stalled shock is large enough, the stalled shock can be revived to produce the successful explosion. This so-called delayed explosion mechanism, or neutrino-driven mechanism, was first

16 proposed by Hans Bethe and Jim Wilson [37]. Matter between the shock and the so-called gain radius undergoes net heating by the outward flow of neutrinos, and net cooling below. For neutrino heating to be successful in powering an explosion, a fluid element must be heated sufficiently while it resides in the heating layer to re-energize the shock [3]. The main energy source of core collapse supernovae explosion is the gravitational binding energy released in the collapse of the iron core from an outer radius of ∼1500 km and central densities below ∼ 1010 g/cm3 to a radius about 10 km and densities above that in atomic nuclei (∼ 3 × 1014 g/cm3). The amount of the gravitational binding energy released is given by

 2  −1 2 GM MNS RNS NS 53 Egrav ∼ ∼ 3 × 10     erg, (2.5) R 1.4M  10km

where MNS is the neutron star mass (typically ∼ 1.4M ) and RNS is its radius (on the order of 10 km). The observed kinetic energy of the material expelled in a supernova explosion is ∼ 1051 erg. Hence, only about 1% of the gravitational binding energy has to be transfered to the accreting matter by neutrinos in order for the neutrino- heating mechanism to successfully produce an explosion [20, 21]. This is why energy deposition by neutrinos plays the crucial role in the neutrino heating mechanism, and the rate of energy deposition per nucleon,q ˙, can be written as

Xn Lνe 1 Xp Lν¯e 1 q˙ = h2 i + h2 i , (2.6) a 2 νe a 2 ν¯e λνe 4πr fνe λν¯e 4πr fν¯e

where the ﬁrst and second terms express the absorption of electron neutrinos (νes)

2 2 and antineutrinos (¯νes), respectively. Lνe and Lν¯e are their luminosity, hνe i, hν¯e i are

their mean square energies, and fνe , fν¯e are their ﬂux factor, which are a measure of

17 νe’s andν ¯e’s anisotropy, deﬁned by 2πc Z L = 4πr2 d dµ 3 µ f, (2.7) νe (hc)3 νe νe νe νe

R d dµ 5 f h2 i = νe νe νe , (2.8) νe R 3 dνe dµνe νe f

1 R d dµ 3 f = νe νe νe . (2.9) R 3 fνe dνe dµνe νe µνe f An accurate calculation ofq ˙ requires an accurate calculation of both the energy spectrum and the angular distribution of the neutrinos [3]. The presence of the neutrino heating region has been conﬁrmed by many simulations [15, 38–40]. Nevertheless, the full physics one-dimensional calculations did not demonstrate a shock revival and subsequent explosion [41–44]. Two-dimensional axisymmetric simulations showed that convection in the heating layer plays an important role by transporting hot gas from the bottom of neutrino-heating region directly to the shock, while downﬂows simultaneously carry cold, accreted matter to the layer of strongest neutrino heating where this gas absorbs more energy from the neutrinos [3]. It has been shown that convection in the heating layer will lead to explosions where the same model with the same neutrino luminosities and RMS energies fail when computed in spherical symmetry [39]. Another important ingredient missing in 1D simulations was indicated by Blondin et al. [4] who discovered the presence of a large-scale low-l hydrodynamic/advective- acoustic instability of the standing accretion- shock (SASI) that leads to time-varying large-scale typically bipolar shock deformations (Figure 2.8) and accretion down- streams that boost the neutrino luminosity. Multidimensional simulations with parameterized neutrino sources have shown that the development of the SASI naturally produces explosions with asymmetries

18 Figure 2.8: Images of the gas entropy (red is higher than the equilibrium value, blue is lower) illustrate the instability of a spherical standing accretion shock. This model has γ = 4/3 and is perturbed by placing overdense rings into the infalling preshock gas. The shock is kept stalled by using a cooling function. Note that with the scaling for a realistic supernova model, the last image on the right corresponds to 300 ms. These simulations are axisymmetric, forcing a reﬂection symmetry about the vertical axis. This ﬁgure is taken from [4] that may explain the observational evidence for asphericities in the core-collapse explosion: pulsar velocities, convective mixing [45, 46], asymmetry of the expanding envelope [47], frequency-dependent linear polarization of the emitted electromagnetic waves [48–50].

19 CHAPTER 3

GRAVITATIONAL WAVES

Gravitational waves (GW) are ripples in the curvature of spacetime that propagate as a wave with the speed of light. Existence of GW was predicted by Albert Einstein in 1916 on the basis of his Theory of General Relativity (GR) [51]. Gravitational waves transport energy and momentum in the form of the radiation. The primary sources of gravitational waves include binary inspirals, spinning neutron stars and black holes, and stellar collapses. All detectable sources of GWs are far away from Earth. It means that our GW observatories are located in a weak-field region, where GR can be linearized and Einstein’s equations become a set of linear wave equations [23,52,53]. Observation of gravitational collapse by GW detectors will provide unique information, complementary to that derived from electromagnetic and neutrino detectors. In contrast with other types of radiation, gravitational waves can propagate from the innermost parts of the stellar core to detectors. With the help of gravitational waves we will be able to see directly the details of the stellar collapse dynamics. The neutrino and the GW signals will finally open for us the mechanism that governs the supernova explosion [22,33,54]. In this section, I will review linearized GR, linear wave solutions and gravitational wave generation in the weak-field, slow-motion limit by essentially-Newtonian matter sources. I will also briefly touch on the detection of gravitational waves. For more details, the reader is directed to textbooks such as [55–57].

20 3.1 GENERAL RELATIVITY

Theory of general relativity is based on the Einstein ﬁeld equation that couple spacetime curvature and energy-matter. The equation can be written in a compact tensor component form as

1 8πG G ≡ R − g R = T , (3.1) µν µν 2 µν c4 µν where Gµν is the Einstein tensor, Tµν the stress-energy tensor, Rµν the Ricci tensor,

µ and R = R µ the Ricci (or curvature) scalar. Despite their simple appearance the equations consist of 10 coupled non-linear partial diﬀerential equations. They only have analytical solutions in the most ideal and highly symmetrical situations, such as for a single non-rotating black hole (Schwarzschild) and a single rotating black hole (Kerr). In any real astrophysical case it is necessary to solve the Einstein equations numerically.

3.2 LINEARIZED THEORY

In regions of spacetime where the curvature is small, gµν may be treated as a ﬁrst order perturbation from the Minkowski metric ηµν = diag( − 1, 1, 1, 1):

gµν = ηµν + hµν with |hµν| 1. (3.2)

Here, hµν is the metric perturbation. Since in linearized theory we only take into account terms linear in hµν, raising and lowering all indices can be done using just

ηµν. In the linear limit, the Ricci tensor becomes

21 λ λ 2 Rµν = Γλµ,ν − Γµν,λ + O h (3.3)

and the connection coeﬃcients are deﬁned by metric

1 Γλ = ηλρ (h + h − h ) + O h2 . (3.4) µν 2 ρν,µ ρν,µ νµ,ρ The ﬁrst-order Ricci tensor will be expressed as

1 R = h − hλ − hλ + hλ , (3.5) µν 2 µν ν,λµ µ,λν λ,µν

µν ∂ ∂ where = η ∂xµ ∂xν is the d’Alambert operator.

Let us assume the Lorentz (harmonic) gauge conditions hµν,ν = 0 and deﬁne a ν ¯ trace h ≡ h ν and a traceless part of hµν :

1 h¯ = h − η h. (3.6) µν µν 2 µν

This assumption yields a very simple form of the linearized Einstein equations

8πG G = h¯ = − T , (3.7) µν µν c4 µν ¯ which are wave equations for hµν.

3.3 WAVE SOLUTION IN VACUUM

In vacuum (Tµν = 0) Equations (3.7) reduce to

¯ hµν = 0 (3.8) and has the well-known plane wave solutions

22 ¯ λ hµν = Aµν exp ikλx . (3.9)

λ Inserting these solutions into Eq. (3.8) yields kλk = 0 which means that gravitational waves travel along null geodesics, i.e. at the speed of light. The polarization tensor

Aµν initially has ten independent components. However, using the Lorentz gauge and the special choice of the gauge transformation (transverse traceless (TT) gauge) reduces the wave ﬁelds to two independent physical degrees of freedom, identiﬁed as polarizations:

  0 0 0 0     0 A11 A12 0   Aµν =   (3.10) 0 A −A 0  12 11    0 0 0 0

3.4 POLARIZATION OF GRAVITATIONAL WAVES

Equation (3.10), a gravitational wave propagating in the positive z-direction, can be rewritten in the following form:

  0 0 0 0     0 h+ h× 0 TT   hij =   (3.11) 0 h −h 0  × +    0 0 0 0 where the dimensionless gravitational wave amplitudes or strains, h+ (“plus ”) and h× (“cross ”) represent two polarizations of a gravitational wave (Figure 3.1). In order to determine the TT components of an arbitrary tensor hij for the plane-wave solution in Eq. (3.9), we have to use a projection tensor Pij to ﬁnd the transverse

23 Figure 3.1: Gravitational wave , which propagates along the +z axis. (a) + polarization; (b) × polarization. Figure taken from [5].

part of hij

ki P = δ − n n n = . (3.12) ij ij i j i k

The transverse component of hij is given by

T l m hij = Pi Pj hlm. (3.13)

TT T To obtain hij , hij must be made traceless (Eq. 3.13).

1 hT T = P lP mh − P P lmh . (3.14) ij i j lm 2 ij lm The plane-wave approximation in Eq. (3.9) is valid for any distance source of gravitational waves. All the interesting sources of GWs satisfy this condition.

24 3.5 GRAVITATIONAL RADIATION IN THE WEAK-FIELD SLOW-MOTION LIMIT

Solutions of the wave equation (3.8) can be given in a form of the time-retarded Green function,

 TT |~r−~r 0| 0 4G Z Tµν t − c , ~r h¯TT (t, ~r) =  d3r0 ,  (3.15) µν c4  |~r − ~r 0| 

in the weak-ﬁeld limit (GM/(c2r) 1) and the slow-motion approximation (|v| c). An expansion in powers of r0/r yields the lowest order so-called quadrupole formula

  2G r h¯TT (t, r) = I¨TT t −  , (3.16) ij c4r ij  c

where the transverse traceless part of the quadrupole moment is given by

  Z 1 ITT = d3x ρ x x − δ r2 . (3.17) ij  i j 3 ij 

The quadrupole part of the total energy radiated in gravitational waves is given by [23]

Z +∞ G ...TT ...TT EGW = 5 dt I ij I ij (3.18) 5c −∞

3.6 GRAVITATIONAL WAVE ASTRONOMY

3.6.1 Estimation of the detectability of gravitational signals from CCSN

Interferomtric detection of gravitational waves is based on the measurement of the change of the physical distance between two masses when a gravitational wave passes

25 through them. With the help of the quadrupole formula (3.14), the characteristic amplitude (the relative displacement) corresponding to gravitational signals from core-collapse supernovae can be estimated as

2G MR2 TT hCCSN ∼ 4 2 , (3.19) c D Tdyn

where D is the distance to the source, M, R, and Tdyn represents the typical mass and radius of the inner core and the dynamical timescale at core bounce, respectively.

The dynamical time Tdyn is given by

1 T ∼ √ . (3.20) dyn Gρ¯

Strain of ravitational signal from CCSN are produced after bounce and a shock for-

13 mation with the typical mass M = 0.5M , densityρ ¯ = 2 × 10 and radius R = 10 km, can be estimated using equations (3.19) and (3.20). For supernova taking place

−21 at our galactic center (D = 10 kpc) they yield Tdyn ≈ 1.2 ms, hCCSN ≈ 3.7×10 . In addition, the typical frequency of the gravitational waves, which can be approximately estimated by the inverse of the dynamical timescale, is expected to be

1 fGW ∼ ∼ 800Hz (3.21) Tdyn

The results presented in this work are in agreement with these estimations.

3.6.2 Gravitational waves detectors

Interferometric detectors such as LIGO [6], VIRGO [58], GEO600 [59], and TAMA300 [60] are the most advanced, currently existing, gravitational wave detectors. We hope to see the ﬁrst direct detection of gravitational waves on the collaboration of these projects. Interferometric detectors measure the gravitational wave ﬁeld by observing its action upon a widely separated set of test masses. The experimental setup is

26 Figure 3.2: Simplified scheme of a light-interferometric gravitational wave detector. Fig- ure taken from [6] very much like that of a two-armed Michelson-Interferometer as shown in Figure 3.2 and was first proposed in the 1970s [61, 62]. At the end of each arm a test mass with a mirror is suspended as freely as possible with a sophisticated pendulum isolation system to minimize the effects of local ground vibrations. A second test mass, with a semi-transparent mirror, is suspended close to the beam splitter in each arm, thus creating two Fabry-Perot cavities (one in each arm) that allow for laser beam recycling [61]. A gravitational wave incident perpendicular to the detector plane with + polarization aligned with the detector arms increases the length of one arm while reducing that of the other. Each arm oscillates between being stretched and squeezed as the wave itself oscillates. The gravitational wave is thus detectable by measuring the separation between the test masses in each arm and searching for this oscillation by means of monitoring changes in the interference pattern of the recombined laser beams in the photo detector. In general, both polarizations, + and ×, of incident gravitational waves influence the test masses. The general detector output h(t) is written as

27 2∆L(t) h(t) = = F (θ, φ, ψ)h (t) + F (θ, φ, ψ)h , (3.22) L + + × × where ∆L is the change in separation of two masses a distance L apart, F+ and F× are detector beam-pattern functions, which depend on the direction of the source (θ, φ) and the orientation ψ of the polarization axes relative to the detector’s orientation (Fig. 9.2) and which have values in the range 0 < |F | < 1. There will be some special

choice of θ, φ, ψ for which F+ = 1 and F× = 0; we shall call this the “optimum source direction and polarization”for the + mode of the waves [61]. In practice, the phase diﬀerence of the two laser beams is monitored by a nulling method in which one keeps the light returning from the two arms always π out of phase so that the output is dark. The error signals of the servo-loop control applied to the mirrors to maintain the dark fringe are directly proportional to the action of an incident gravitational wave. The sensitivity of a light-interferometric gravitational wave antenna depends on the arm length and the amount of light energy stored in the arms [214]. To exclude acoustic disturbances and ﬂuctuations in the local index of refraction, the interferometers operate in ultra-high vacuum at pressures below 10−13 bar. The main sources of noise competing with physical signals are [63]:

• Seismic Noise: External mechanical vibrations lead to displacements of the mirrors and are typically many orders of magnitude larger than an astrophysical signal. Seismic noise is the limiting noise below ∼ 10 Hz and a combination of active filters (piezoelectric actuators), passive filters (alternate layers of steel and rubber) and a multi-stage pendulum suspension of the optical components allow effective filtering above 10 Hz.

• Thermal Noise: Brownian motion of the mirrors and excitation of violin

28 modes of the suspension can mask gravitational waves. The pendulum suspensions have thermal noise at a few Hz and internal vibrations of the mirrors have natural frequencies of several khz. To ensure that most of the vibration energy of both kinds of oscillations is conﬁned to a small bandwidth around their natural frequencies, extremely high Q material and low temperatures around 1K are desirable. However, present-day detectors operate at room temperature. Thermal noise is limited between 50250Hz.

• Photon shot noise: Photons arrive and make random ﬂuctuations in the light √ intensity that could be misinterpreted as n gravitational wave information. As a random process, the measurement error improves with n, where n is the number of photons. The shot noise related error displacement is given by [63]

λ δlshot ∼ √ (3.23) 2π N

To measure at a frequency f, one has to make at least 2f measurements per second, so one can accumulate photons for a time 1/2f. With light power P ,

one gets N = P/(hc/λ)/(2f) photons. In order that δlshot should be below 10−16m, one needs high light power, far beyond the output of any continuous laser. Photon shot noise is the principal limitation to sensitivity for frequencies above ∼250 Hz. Present-day detectors operate with lasers with output on the order of 10 W and the necessary high light power is obtained via resonant cavities and beam recycling [63]

• Quantum effects: Besides photon shot noise, additional quantum effects like zero-point vibrations of mirror surfaces affect the detector sensitivity. For present-day detectors these quantum noise effects are small and limit the sen-

29 sitivity to h ∼ 10−25 [63].

• Gravity gradient noise: Changes in the local gravitational ﬁeld on the timescale of the measurements lead to tidal forces on a gravitational wave detector that cannot be screened out. Environmental noise comes from human activities (e.g., lumbermen cutting trees near a detector), and seismic waves, the surf of the sea, and changes in air pressure/density. Gravity gradient noise is the dominant noise component in the frequency band below ∼ 1 Hz and is the primary reason why the detection of gravitational waves in that frequency band must be carried out in space [63].

At this point, six large laser-interferometric gravitational wave detectors are in operation.

• LIGO: The MIT/Caltech Laser Interferometer Gravitational Wave Observa- tory operates, since 2001, with two 4 km interferometers located at Hanford, Washington and Livingston, Louisiana and one 2 km interferometer located in Hanford, Washington. In 2009, LIGO reached its design sensitivity of h ∼ 10−22 Hz−1/2 within a frequency band of ∼ 60 − 1000 Hz [6].

• GEO600: GEO600 is a joint British-German detector with a 600 m arm length that operates, since 2001, near Hannover, Germany under the auspices of the Albert-Einstein-Instituts section at Hannover University. GEO600 incorporates more modern and improved detector design to make up for its short arm length. In 2009, GEO600 reached its design sensitivity of h ∼ 10−21 Hz−1/2 [59]

• TAMA300: TAMA300 operates, since 1999, in Tokyo, Japan. The 300 m arm length interferometers sensitivity is ∼ 10−20 Hz−1/2 within a frequency band of 300 Hz – 8 kHz [60].

30 Figure 3.3: Model curves of optimal sensitivity of Advanced LIGO for diﬀerent type of GW sources. The models take into considerations the principal contributors of the noise. Figure taken from [6]

• VIRGO: VIRGO is French-Italian 3 km interferometer that has started taking data in 2006. It incorporates advanced interferometer design features, including improved test mass suspensions, allowing for sensitivities similar to the LIGO detectors down to frequencies of ∼ 10 Hz [58].

Characteristic Strain

A characteristic gravitational wave is deﬁned by Flanagan & Hughes [64], as shown in Figure 3.3, by

31 s 2G dE (f) h = GW (3.24) char π2c3r2 df where the spectral gravitational wave energy density is

dE (f) c3 (2πf)2 2 GW = A˜ (f) (3.25) df G 16π 20

and ∞ Z ˜ i 2πft A20 (f) = e A20 (t) . (3.26) −∞ Flanagan & Hughes have also introduced the signal-to-noise ratio for an optimally

oriented detector as a function of hchar

Z ∞ 2 2 df hchar(f) (SNRopt) = 2 (3.27) 0 f hrms(f)

with given detector rms noise strain hrms(f) (Figure 3.3) The second generation of gravitational wave detectors, such as the advanced LIGO, the LCGT, and the advanced Virgo, are under construction now. This is a major upgrade, with the goal of increasing the sensitivity (h ∼ 10−23 Hz−1/2) by about one order of magnitude, with respect to ﬁrst generation of GW detectors in the whole detection band. The upgraded detectors are expected to see many events every year. A new era of gravitational wave astronomy may begin with the scientiﬁc run in 2015 [6,58].

32 CHAPTER 4

CHIMERA CODE

The mechanism of supernova is such a complex and non-linear problem, involving very diﬀerent areas of macro and micro physics (hydrodynamics, gravity, equation of state, transport processes, strong and weak interactions, nucleosynthesis), that one cannot expect to solve the supernova problem without numerical simulations. The complexity of the supernova problem imposes great technical challenges on the numerical simulations. A typical supernova explosion energy is about 1051 ergs which is two orders less than the energy released by the core in neutrinos of all ﬂavors. Neutrino interactions with the stellar core will either power the explosion itself or play a major role in the explosion dynamics. An inaccurate treatment of neutrino transport can qualitatively change the results of a simulation. Since neutrinos are trapped within the neutrinosphere and then propagate freely the transport scheme must be accurate in both regimes plus the all-important intermediate regime where the critical neutrino energy deposition occurs. Neutrinos interact with matter in a variety of energy-dependent ways, and this demands that both the neutrino transport and the interactions receive a full spectral implementation, rather than having the neutrino spectrum prescribed. The angular distribution of the neutrinos is also crucial for accurate simulations. Supernova simulations must be carried out in two, or even three, spatial dimensions. The nuclear abundances should be evolved in regions where nuclear statistical equilibrium (NSE) cannot be maintained. This can be used to compare outcomes of

33 the supernova simulations with the observational data. General relativistic effects must be incorporated, as they influence the size of the neutrino heated region, the rate of matter advection through this region, and the neutrino luminosities and RMS energies [65]. They will affect the dynamics of explosion. Therefore, to perform the simulations with sufficient realism, any CCSN code has to include highly accurate and efficient finite difference schemes for hydrodynamic evolution, neutrino transport and a nuclear reactions network as well as the effects of general relativity. There are only few codes that satisfy some of these criteria [3, 13, 66–70]. The results presented in this work were obtained in series of the core- collapse simulations with CHIMERA code. This code is developed by the FAU – ORNL – NCSU – UCSD collaboration [71]. CHIMERA is designed to simulate core-collapse supernovae in 1, 2, and 3 spatial dimensions and with realistic spectral neutrino transport from the onset of collapse to about 1 second after bounce with the present generation of supercomputers. It conserves total energy (gravitational, internal, kinetic, and neutrino) to within 0.5 B, given a conservative gravitational potential. The code currently has five main components: hydrodynamics, neutrino transport, effective GR potential, a nuclear reaction network and a sophisticated equation of state [72]. In the next sections I will briefly describe the architecture of the code.

4.1 HYDRODYNAMICS IN CHIMERA

The hydrodynamics of core collapse supernova are fraught with unique challenges that are not present in other astrophysical problems. The matter is described by complex equations of state for hot, dense matter at both super- and subnuclear densities. The

34 range of densities is over ten orders of magnitude in the core of progenitor stars. For simplicity of implementation, supernova codes usually use explicit hydrodynamics methods, with either a Newtonian or a general relativistic formulation. These methods are easy to implement on parallel architectures. Explicit algorithms require only nearest neighbor communication among processors and are well suited to scalable architectures of the current supercomputers. Implicit algorithms have been avoided since they require computationally expensive solutions of large systems of coupled linearized equations. Regardless of which approach is used, the hydrodynamic portion of the problem requires the solution of the following equations, expressed here in the Newtonian formalism:

∂ρ + ∇ · (ρv) = 0 (4.1) ∂t

  ¯ ∂(ρYe) X Z Se Se + ∇ · (ρY v) = −m d  −  (4.2) ∂t e b   f

∂E X Z + ∇ · (Ev) + P ∇ · v = −m d S + S¯ (4.3) ∂t b e e f

" # ∂(ρv) X Z + ∇ · (ρvv) + ∇P + ρ∇Φ + ∇ · d P + P¯ . (4.4) ∂t e e f Equation (4.1) is the continuity equation for mass, where ρ is the mass density and v is the matter velocity, and where these quantities, and those in the following equations, are understood to be functions of position ~x and time t. Equation (4.2) expresses the evolution of electric charge, where Ye is the ratio of the net number electrons over positrons to the total number of baryons. In the presence of weak interactions, the

35 right hand side is non- zero to account for reactions where the number of electrons ¯ can change. Here, Se is the net emissivity of a neutrino flavor (of energy ) and Se is its antineutrino counterpart. This expression is integrated over all neutrino energies and summed over all neutrino flavors f . The mean baryonic mass is given by mb. Evolution of the internal energy of the matter is given by the gas-energy equation (4.3), where E is the matter internal energy density and P is the matter pressure. Again, the right hand side of this equation is non-zero whenever energy is transferred between matter and neutrino radiation as a result of weak interactions. We note that it is also possible to substitute Eq.(4.3) for an expression for the evolution of the total matter energy (internal plus kinetic plus potential). Finally, Eq. (4.4) expresses gas- ¯ momentum conservation, where Φ is the gravitational potential, and Pe, and Pe, are radiation-pressure tensors for each energy and flavor of neutrino and its antineutrino, respectively. These equations must be discretized to be solve in a computational framework. CHIMERA uses a modified version of the VH-1 code [73], which combines Eu- lerian and Lagrangian hydrodynamics. VH-1 is based on the Piecewise Parabolic Method (PPM) [74]. Being third order in space (for equal zoning) and second order in time 4.1, the code is well suited for resolving shocks, composition discontinuities, etc. with modest grid requirements. It is implemented in Fortran + MPI and has been optimized for parallel architectures. The code has been validated on a wide variety of test problems [75]. The PPM algorithm, employed in VH-1, offers supe- rior shock resolution capabilities. To avoid the odd-even decoupling and carbuncle phenomenon for shocks aligned parallel to a coordinate axis, CHIMERA incorporates the local oscillation filter method of Sutherland et al. [76], which subjects only a minimal amount of the computational domain to additional diffusion and also the geometry corrections of Blondin & Lundqvist [77] in the hydrodynamics module to

36 Figure 4.1: Convergence factor as obtained from the convergence study with the diﬀerent angular resolutions h (64 angular zones), 2h (32 angular zones), 4h (16 angular zones) and ﬁxed radial resolution (128 zones). Convergence factor of 3 corresponds to 1.6 order algorithm. The possible reason of the lower convergence is a presence of a sharp discontinuity at the shock surface. One can expect to see the second order convergence higher resolutions in both space directions.

37 avoid spurious oscillations along the coordinate axes. A moving radial grid option, where the radial grid follows the average radial motion of the ﬂuid, makes it possible for the core infall phase to be followed with good resolution. Following bounce, an adaptive mesh redistribution algorithm keeps the radial grid between the core center and the shock structured so as to maintain approximately constant ∆ρ/ρ.

4.2 GRAVITY IN CHIMERA

As we already emphasized in Chapter 2, gravity plays an essential role during all stages of a core collapse supernova. First of all, gravity is the force that initiates the collapse of the stellar core. Furthermore, the energy required for supernova explosions comes from the enormous amount of gravitational binding energy released during the formation of the proto-neutron star. So, it is important that the algorithm used in CCSN simulations calculates the gravitational potential eﬀectively and accurately through these stages. Gravity in CHIMERA can be modeled by both the Newtonian potential and the modiﬁed Newtonian potential [78]. The integral form of the Poissons equation in spherical coordinates

Z ρ(r~0) 3~0 Φ(~r) = −G d r (4.5) ~0 ~r − r allows s to expand the integrand in terms of spherical harmonics, Ylm(θ, φ). The algorithm is described by M¨uller& Steinmetz [79] in detail. We summarize brieﬂy the key steps of the method. Expanding Eq. (4.5), we get

38 ∞ l X 4π X Z Φ(r, θ, φ) = −G Y (θ, φ) dΩ0 Y (θ0, φ0) (4.6) 2l + 1 lm lm l=0 m=−l   1 Z r Z ∞  dr0 r0l+2ρ(r0, θ0, φ0) + rl dr0 r01−lρ(r0, θ0, φ0) .  l+1  r 0 r

The advantage of the method is that we only need to calculate the local zone integrals.

Z φk Z θj Z ri lm ∗ l+2 Aijk = sin θ dθ dφ Ylm(θ, φ) dr r ρ(r, θ, φ) (4.7) φk−1 θj−1 ri−1 Z φk Z θj Z ri lm ∗ 1−l Bijk = sin θ dθ dφ Ylm(θ, φ) dr r ρ(r, θ, φ) φk−1 θj−1 ri−1

Now we just add up these integrals over all zones of the spherical grid (Nr ×Nθ ×Nφ).

Nr Nθ Nφ lm X X X lm Cn = Aijk (4.8) i=1 j=1 k=1

Nr Nθ Nφ lm X X X lm Dn = Bijk. i=1 j=1 k=1 The potential at any grid point can then be generated by a sum of the spherical harmonics weighted by these coeﬃcients. Note that the only global communication

lm lm occurs in the summation that produces Cn and Dn . As a result, the algorithm requiring only a single global sum across processors is both eﬃcient and highly scalable. In practice, one assumes the density is a slowly varying function of (θ, φ), such that the angular zone integrals in Eq. (4.8) can be pre-computed assuming a constant density within each zone [7]. Most of multidimensional core-collapse supernova simulations by other groups were done in the framework of Newtonian gravity [66–70]. However, it has been

39 shown by Bruenn et al. [65] that the effects of general relativistic hydrodynamics are important and cannot be neglected in quantitative models because of the increasing compactness of the postcollapse core and the changing the neutrino emission (luminosities and spectra) during the shock reheating epoch. Additionally, important observable effects such as the time dilation and the gravitational redshift for neutrino transport can be take into account only in GR simulations. In CHIMERA, the main GR effects mentioned above are reproduced by a general relativistic corrections to the spherical component of the potential suggested by Marek et al. [78], which is known as “effective GR potential”.

4.2.1 Derivation of eﬀective GR potential

The spherically symmetric spacetime metric can be written as

ds2 = −e2φ(r)dt2 + e2λ(r)dr2 + r2dΩ2. (4.9)

For a self-gravitating fluid it is desirable that an effective relativistic potential repro- duces the solution of hydrostatic equilibrium according to the Tolman-Oppenheimer- Volkoff (TOV) equation [80,81]

  Γ2 ∂p m + 4πr3p D U = −   − , (4.10) t + p ∂r r2

q −φ ∂r 2 2m where we deﬁne U as U ≡ Dtr = e ∂t , Γ = 1 + U − r is the Lorentz factor , and m = R ΓdV.

1 + p m + 4πr3p ∇ Φ = (4.11) r GR Γ2 ρ r2

40 4.2.2 Implementation

There is an ambiguity in the definition of the empirical potential form. It depends on how the Newtonian limit is taken. Marek et al. [78] performed series of test runs with the different modifications of the potential and found, as the best choice to reproduce the infall velocities and the compactness of PNS in full GR is

  3 1 ρ(1 + e) + p m + 4πr (p + pν) ∇ Φ =   , (4.12) r TOV Γ2  ρ  r2

where the Lorentz factor is deﬁned with the local radial velocity only

r 2m Γ = 1 + (v )2 − . (4.13) r r The TOV mass is given by

Z v F m ≡ 4πr2 ρ(1 + e) + e + r ν dr. (4.14) ν Γ Here ρ is the rest-mass density, e = ρ is the internal energy density with being the

specific internal energy, p is the gas pressure, and pν, eν,Fν are the neutrino pressure, the neutrino energy density, and the neutrino flux, respectively. In order to calculate the effective relativistic potential for multi-dimensional flows we substitute the “spherical contribution”Φ(r) of Newtonian gravitational potential by the TOV potential

∞ X Newton ΦGR(r, θ, φ) = ΦTOV (r) + Φl (r)Ylm(θ, φ) (4.15) lm

41 Figure 4.2: Time evolution of the central density ρc for Model AB-GR ( black solid line), Marek et al. (black dotted line), CHIMERA eff. potential (Coordinate time) (green solid line), CHIMERA eff. potential (Proper time) (green dashed line), CHIMERA Newtonian potential (Coordinate time) (pink solid line), The left panel shows the collapse phase, while the right panel shows the post-bounce evolution. Note the different axis scales in both panels.

4.2.3 Tests of the eﬀective GR potential

The implementation of the effective GR potential in CHIMERA has undergone a series of tests. Figure 4.2 shows the central density as a function of time for the collapse (left panel) and for the subsequent postbounce phase (right panel). We compare results of the CHIMERA simulations with the effective potential with those of AGILE-BolzTran (AB-GR) simulations. AGILE-BolzTran is 1D full GR hydrodynamics code perform- ing an multigroup Boltzmann neutrino transport [82]. The “central”density is the density value at the center of the innermost grid zone of the AB-GR simulation. Be- cause of different numerical resolutions it was necessary to interpolate the CHIMERA

42 Figure 4.3: Comparison of the radial velocity profile at 100 ms between CHIMERA simulation with the effective GR potential and 1D full GR code AGILE-BolzTran simulations. results to this radial position. There are no significant differences between the full GR calculation and effective potential calculations during the collapse (left panel of figure 4.2). On the other hand, we can see the discrepancy in the postbounce behavior of the central density between CHIMERA implementation of effective potential and the one in Marek et al [78]. The main reasons for this discrepancy come from differences in initial models used in both codes and in the treatment of neutrino transport. In spite of the slight post-bounce deviation of the effective potential density function from the GR density function, the radial velocity profiles are in a good agreement (Figure 4.3) We also looked at radial pulsations of protoneutron stars in the effective gravity. It has been found that the fundamental mode in this case deviates from the one in GR simulations (Figure 4.4). This deviation may not be significant for evolution of most of the post-bounce epoch but the part of the gravitational wave signal that is produced by protoneutron

43 Figure 4.4: Normalized eigenfrequency of fundamental radial mode of non-rotating relativistic stars (n = 1 polytrope) plotted as a function of compactness parameter M/R: ”original eﬀective“ means original form of the eﬀective potential, ”Γ = 1 in Equil“ means applying Lorentz factor Γ only in computing equilibrium. In ”Marek et al. Case (A)“, Lorentz factor is introduced both in equilibrium and in perturbation.

44 star pulsations may be aﬀected.

4.3 NEUTRINO TRANSPORT

This component is the most diﬃcult to implement in a supernova model and the most time-consuming computationally. This is because supernova neutrinos cannot, in general, be described by an equilibrium distribution function. A solution requires a complete phase-space description of each neutrino’s position and momentum. To obtain such a solution, one must solve the six-dimensional Boltzmann Transport Equation or some reasonable approximation thereof. This extra dimensionality easily leads to the transport calculation completely dominating a simulation in terms of computer memory, execution time, and I/0 requirements. The Boltzmann Transport Equation (BTE) can be expressed in terms of the radiation intensity, I = I(, ~x, Ω, t), where is the energy of a neutrino, ~x its position, and Ω the solid angle into which the neutrino radiation is directed. In terms of I, the Newtonian BTE can be expressed as

  1 ∂I X ∂I ∂f + Ω~ · ∇I + a =   , (4.16) c ∂t i ∂p  ∂t i i coll th th where ai is the i component of the matter acceleration and pi the i component of the momentum of the neutrino. The right hand side of Eq. (4.16) combines together the contributions from all interactions that a neutrino might experience and is collectively referred to as the collision integral. Mezzacappa and Bruenn implemented the “full”solution to the BTE for 1D supernova simulations [83]. For multi-dimensional models, the full solution of the BTE becomes yet more chal- lenging to implement and more time-consuming to compute. Approximate transport moment methods are applied for 2D and 3D models.

45 CHIMERA uses a multi-group flux-limited diffusion (MGFLD) scheme. The MGFLD combines separate equation for each neutrino energy (multi-group) and a finite series of angular moments of the BTE (flux limited). The flux limiter has been tuned to reproduce results of a general relativistic Boltzmann transport solver to within a few percent [84]. Neutrino transport is performed by a “ray-by-ray- plus”approximation [85] whereby the lateral effects of neutrinos such as lateral pressure gradients (in optically thick conditions), neutrino advection, and velocity corrections are taken into account, but transport is performed only in the radial direction (Figure 4.5). All O(v/c) observer corrections have been included. The transport solver is a fully implicit algorithm for numerical solution of the Boltzmann equation for four neutrino

flavors (νe, ν¯e, νµ,τ , ν¯µ,τ ) simultaneously, allowing for neutrino-neutrino scattering, pair exchange, and different neutrino and antineutrino opacities. The PPM method has been directly applied to both the spatial advection of neutrinos in both the radial and lateral directions and the energy “advection ”associated with neutrino frequency shifts. The neutrino opacities employed for the simulations are the standard ones described in [36] with the isoenergetic scattering of nucleons replaced by the more exact formalism of Reddy et al. [86], which includes nucleon blocking, recoil, and relativistic effects, and with the addition of nucleon-nucleon bremsstrahlung [87], with the kernel reduced by a factor of five in accordance with the results of [88].

4.4 NUCLEAR REACTION NETWORK

The nuclear composition in the non-NSE regions of these models is evolved by the thermonuclear reaction network of Hix & Thielemann [89]. This is a fully implicit general purpose reaction network, which contains a system of diﬀerential equations

46 Figure 4.5: Schematic of CHIMERA’s assumed geometry. The hydrodynamics is evolved in the three-dimensional space deﬁned by r, θ, φ. At each point along each r, a neutrino phase space is evolved as well, spanned by 2 neutrino propagation angles, Θ and Φ, and a neutrino energy, . The heart of the ray-by-ray-plus approximation lies in the assumption that each ray, r, can, for the most part, be decoupled from neighboring rays during the transport step, requiring only integral corrections (e.g. lateral pressure gradients). We assume the imposed boundary conditions on each radial transport solution mimic the eﬀects of neighboring rays. The additional degrees of freedom provided by the nuclear network are not shown. Diagram from [7]

47 for the nuclear abundances of the form:

i ˙ X i X i X 2 2 Yi = Nj λjYj + Nj,kρNAhj, kiYjYk + ρ NAhj, k, liYjYkYl, (4.17) j j,k j,k,l

Yi = ni/ρNA is the nuclear abundance, where ni is the number density, ρ is the mass density, NA is Avogadro’s number. The N s provide for proper accounting of

i i Qnj,k i numbers of nuclei and are given by: Nj = Ni, Nj,k = Ni/ m=1 Nm!, and Nj,k,l =

Qnj,k,l Ni/ m=1 Nm!. The Ni’s can be positive or negative numbers that specify how many particles of species i are created or destroyed in a reaction, while the denominators, including factorials, run over the nj,k or nj,k,l different species destroyed in the reaction and avoid double counting of the number of reactions when identical particles react. The first term of Eq. (4.17) describes changes due to the reactions involving a single nucleus, which include decays, electron captures, photodisintegrations and so on. λj is the one particle interaction rate. The second and third term describe changes due to two and three-body reactions, respectively [90]. For a set of nuclear abundances ˙ Y~ , one can calculate the time derivatives of the abundances Y~ using Eq. (4.17). The desired solution is the abundance at a future time, Y~ (t + ∆t), where ∆t is the network timestep. Most of the past and present nucleosynthesis calculations use the simple finite difference scheme:

~ ~ Y (t + ∆t) − Y (t) ˙ ˙ = (1 − Θ) Y~ (t + ∆t) + ΘY~ (t) . (4.18) ∆t For the stiff set of non-linear differential equations which form most nuclear reaction networks, a fully implicit treatment is generally most successful [91]. Solving the fully implicit version of Eq. (4.18) is equivalent to finding the zeros of the set of equations

~ ~ Y (t + ∆t) − Y (t) ˙ Z~ (t + ∆t) ≡ − Y~ (t + ∆t) = 0. (4.19) ∆t

48 This is done using the Newton-Raphson method, which is based on the Taylor series expansion of Z~ (t + ∆t), with the trial change in abundances given by

 −1 ~ ∂Z (t + ∆t) ˙ ∆Y~ =   Y~ (t + ∆t) = 0, (4.20) ∂Y~ (t + ∆t) where ∂Z~/∂Y~ is Jacobian of the system of equations. In the current models, this fully implicit general purpose reaction network employs reactions linking only the 14 alpha nuclei from 4He to 60Zn (see Figure 4.6). Data for these reactions is drawn from the REACLIB compilations [92]. The advection of material across an NSE-nonNSE interface in either direction performed as detailed in [72]. An additional iron-like nucleus is included to accommodate a neutron-rich freezeout, which the alpha network is unable to follow, as all included nuclei have equal numbers of neutrons and protons.

4.5 EQUATIONS OF STATE

The equation of state (EOS) of Lattimer & Swesty [93] is currently used in CHIMERA for matter in NSE above 1.7 × 108 g/cm3 . The Lattimer-Swesty EOS is based on a phenomenological, finite-temperature compressible liquid drop model. It also includes surface effects as well as electronpositron and photon contributions. The LS EoS assumes a nuclear symmetry energy of 29.3MeV, nuclear compressibility modulus K ∼ 180 – 220 MeV and density variations 1.66 × 107 g/cm3 < ρ < 1015g/cm3. In between the data points of the table, bisection is applied. Below density 1.7 × 108 g/cm3, matter in non-NSE is described by 4 nuclei (neutrons, protons, helium, and a representative heavy nucleus) in a highly modified version of the EOS of Cooperstein [94], which is based on the ideal gas approximation for the gas of nucleons and nuclei.

49 Figure 4.6: A portion of the nuclear N-Z plane important for supernova nucleosynthesis. The color map gives the abundance of each species at the indicated density and temperature (roughly 6 × 105 g/cm3 and 4.7 billion K, respectively). The locations of the current 14 isotope network species are marked with αs. The extents of the 150 isotope network are circumscribed by the dashed line, exhibiting the good, but not complete, coverage for the chosen conditions. Figure taken from [7].

50 Figure 4.7: The distribution of tracer particles for the 12M model: left panel: at t = 235 ms, center panal: at t = 240 ms, right panel: at t = 440 ms elapsed time. Figure from [8].

4.6 TRACER PARTICLE METHOD

While Eulerian schemes are preferred for regions with violent turbulence to avoid self- intersecting in the grid, hey have a disadvantage: the history of ﬁeld variables for a given parcel of material, crucial for nucleosynthesis, is lost [95]. To compensate for this loss, and to allow post-processed nuclear network computations, the tracer (or test) particle method [95,96] has been implemented in CHIMERA [8]. The tracer particles are equally distributed on the spherical grid (40 particles/row x 125 rows) at the pre- collapse phase and follow the ﬂow in the course of the Eulerian simulation (Figure 4.7), recording their temperature and density history by interpolating the corresponding quantities from the underlying Eulerian grid [8]. Each particle is assigned a constant mass (1/5000 of the progenitor mass) and the GW signal it produces is calculated taking the quadrupole integral. Comparing the GW corresponding to a given group of tracers with the signal produced by the bulk matter motion allows us to identify

51 what part of the ﬂuid generates a speciﬁc GW feature.

52 CHAPTER 5

GRAVITATIONAL WAVE ANALYSIS

In this chapter we describe the code written for the post-processing extraction of gravitational wave signals from data obtained in the simulations of axisymmetric core-collapse supernova with CHIMERA.

4 Stellar collapse with typical values of GM/c r < 0.04 and vmax/c < 0.2 satisﬁes the weak-ﬁeld slow-motion conditions (see §3.5). Thus, GW extraction subroutines are based on the quadrupole formalism. I wrote these codes in close collaboration withe Dr. Shin Yoshida [97]

5.1 MASS QUADRUPOLE OF GRAVITATIONAL WAVE

Taking into account an accuracy of the supernova simulations we restrict ourselves to consider the lowest order terms in the retarded expansion of mass-quadrupole formula (3.16)[98]. The transverse-traceless part (TT) of the gravitational strain is given by

m=2 2 1 X d r hTT = I t − f 2m, (5.1) ij r dt 2m c i,j m=−2 r where mass quadrupole I2m t − c is deﬁned as

16πG√ Z I = 3 τ Y ∗ r2dV, (5.2) 2m 5c4 00 2m 1 where τ00 is the (00)-component of linearized stress-energy tensor . In the weak-

ﬁeld limit, τ00 is approximated by the rest mass density of matter. Tensor spherical 1Originally it contains not only the matter contribution but also gravitational the pseudo tensor, which can be omitted in the linearized theory.

53 harmonics f `m(θ, φ)(θ, φ are angular coordinate of observer’s frame) are explained in Appendix A.

The amplitude factor A2m of the gravitational ﬁeld is

2 d I2m A = . (5.3) 2m dt2 Optimal design of gravitational wave detectors requires some knowledge of the expected waveforms, the corresponding frequency spectra, and the total energy emitted by possible sources [61]. Thus, any extraction method should decrease the numerical noise as much as possible. Most of numerical diﬀerentiation methods amplify numer-

ical noise built-in the simulation data. To avoid this, A2m is usually computed by

reducing the order of time derivatives of I2m:

dN2m dI2m A = , where N = . (5.4) 2m dt 2m dt Following Nakamura & Oohara [99], Blanchet et al. [100] and Finn & Evans [101], the quadrupole signal can be expressed in terms of a volume integral, depending only on the density, velocity and the gradients of gravitational potential

d Z Z ∂ρ ρr2Y ∗ dV = r2Y ∗ dV. (5.5) dt 2m ∂t 2m Using the continuity equation (4.1) in the integrand in Eq. (5.5), one can replace

∂tρ. Integrating by parts and omitting the surface contribution, we ﬁnd the resulting integrand

√ 16π 3G Z 2π Z π Z ∞ 0 3 N2m = 4 dϕ dϑ dr r c 0 0 0 ∂ ∂ 2ρvrˆY ∗ sin ϑ0 + ρvθˆ sin ϑ0 Y ∗ + ρvϕˆ Y ∗ (5.6) 2m ∂ϑ0 2m ∂ϕ0 2m

54 where vaˆ is the component of velocity in the orthonormal frame. Further reduction of the time derivative is conventionally done by using the momentum (Euler) equation (4.4)[99, 100]. However, it has to be carried out carefully.

The Euler equation includes stress terms. Therefore, in order to replace ∂t(ρv) with balancing stress terms, we need to take into consideration all possible contributions in the stress terms, such as pressure, gravity, anisotropic neutrino force acting on the fluid, effective viscosity that may present in finite differencing of themomentum equation, etc. To avoid this issue, we have decided to compute N2m for all the time steps and numerically evaluate dN2m/dt to obtain A2m. Numerical algorithms for computing first order derivatives introduce far less numerical noise than the ones for higher order derivatives. Actually, this approach was suggested in the work of Finn & Evans [101] as one of the alternatives to evaluating Eq. (5.1).

5.2 GRAVITATIONAL WAVES PRODUCED BY ANISOTROPIC NEUTRINO EMISSION

Besides aspherical mass motion, any other sources with non-zero quadrupole moments will produce gravitational radiation. One of these sources is the anisotropic radiation of neutrinos from a hot PNS. The theoretical derivation of the GW signal produced by a distant anisotropic point source of neutrino was first done by Epstein [102]. Müller& Janka [103] were the first authors who implemented this formalism. Kotake et al. [9] improved it and made it more suitable for numerical evaluations of GW signals.

TT The transverse-traceless part of the gravitational strain hij from neutrinos is given by [103]

55 Figure 5.1: Schematic ﬁgure of coordinate frames.

Z t−R/c Z TT TT 4G 0 0 (ninj) dL 0 0 0 hij = 4 dt dΩ (ϑ , ϕ , t ). (5.7) c R −∞ Ω 1 − cos θ dΩ

Here R is the distance from the source to the observer, vector ni the direction of neutrino emission whose components are given with respect to observer’s frame. TT is the transverse-traceless part of the second rank symmetric tensor with respect to the observer’s z-axis (z-direction is deﬁned as the one connecting the source and the

TT observer). The (ninj) term is explained in Appendix B. (θ and φ) are the direction angles of neutrino emission with respect to the O-frame. The other factor in the integrand, dL/dΩ, is the “direction dependent neutrino luminosity”given in S-frame (see Figure 5.1). In the case of axisymmetry, both components of the gravitational wave signal vanish for an observer on the symmetry axis and the GW signal with the maximum amplitude will be detected by an observer in the equatorial plane [103]. For such observer, the gravitational ﬁeld is given by

56 Figure 5.2: Angular dependence of Ψ in Eq. (5.10). Note that the angle is measured from the symmetry axis. Figure is taken from [9].

Z t−R/c Z TT 2G 0 0 0 0 dL 0 0 0 hν,e = 4 dt dΩ Ψ(ϑ , ϕ ) (ϑ , ϕ , t ), (5.8) c R −∞ 4π dΩ where

cos2 ϑ0 − sin2 ϑ0 sin2 ϕ0 Ψ(ϑ0, ϕ0) = (1 + sin ϑ0 cos ϕ0) . (5.9) cos2 ϑ0 + sin2 ϑ0 sin2 ϕ0 We can simplify Ψ (ϑ0, ϕ0) integrating by ϕ0 [9]

Z 2π 1 + sin ϑ0 cos ϕ0 Ψ(ϑ0) = sin ϑ0 −π + dϕ0 = π sin ϑ0 (−1 + 2 |cos ϑ0|) . 2 0 2 0 0 1 + tan ϑ sin ϕ (5.10) This is the ﬁnal equation form that we use in our code to compute the gravitational wave signal produced by anisotropic neutrino emission. Figure 5.2 shows Ψ as a function of neutrino beam direction ϑ0.

5.2.1 Deﬁnition of angles

Following the notation used in M¨uller& Janka [103] and Kotake et al. [20], we setup two sets of coordinate frames: one for source coordinate (S-frame) and the other for

57 observer coordinate (O-frame) (see Figure 5.1). The source coordinate is the one used in a typical radiation-hydrodynamics simulation. To specify where the observer is located in terms of source coordinates (S-frame, denoted by (x0, y0, z0) in Figure 5.1), we use ϑ0 and ϕ0 with the usual definition of spherical polar coordinates. The relative orientation of the observer’s frame (O-frame, denoted by (x, y, z)), specified by the z-axis, is pointing from source toward observer and the y-axis is always parallel to the source’s x − y plane. To specify the direction of neutrino (n) flying from the origin of S-frame, we need another set of angles (ϑ, ϕ). Let unit vector ~n be a direction of neutrino emission. Then,

0 0 0 0 0 ~n = sin ϑ cos ϕ ~ex0 + sin ϑ sin ϕ ~ey0 + cos ϑ ~ez0 , (5.11)

where ~ex0 ,~ey0 ,~ez0 are unit vectors in S-frame. On the other hand, the same direction of the neutrino emission can be expressed in the O-frame

~n = sin θ cos φ xˆ + sin θ sin φ yˆ + cos θ z.ˆ (5.12)

Herex, ˆ y,ˆ zˆ are unit vectors in O-frame. The polarization of the GW is deﬁned with respect to the O-frame, while its amplitude is computed in S-frame. This means that we need to establish a relation between (ϑ0, ϕ0) and (θ, φ). The GW strain is written as integrals containing both (θ, φ) and (ϑ0, ϕ0)(5.7). This expression can be written in terms of the S-frame angles (ϑ0, ϕ0) only. In order to do this, we set Eq.(5.11) equal to Eq.(5.12). For instance,

0 the components ofx ˆ are obtained by rotating ~ex0 around the y -axis by ϑ and then rotating around the z0-axis by ϕ

xˆ = Rz0 [ϕ]Ry0 [ϑ](1, 0, 0). (5.13)

58 0 Here Rz0 and Ry0 represent the axis rotation matrices for a rotation about the z -axis and the y0-axis, respectively. From Eq.(5.12) we get

  cos ϕ cos ϑ sin θ cos φ − sin ϕ sin θ sin φ + cos φ sin φ cos θ     ~n = sin ϕ cos ϑ sin θ sin φ + cos ϕ sin θ sin φ + sin φ sin φ cos θ  . (5.14)     − sin ϑ sin θ cos φ + cos ϑ cos θ

Now we make the components of ~n (Eq. (5.14)) equal to those in Eq. (5.11), to obtain an implicit relation between (ϑ0, ϕ0) and (θ, φ).

sin ϑ0 cos ϕ0 = cos ϕ cos ϑ sin θ cos φ − sin ϕ sin θ sin φ + cos φ sin φ cos θ (5.15)

sin ϑ0 sin ϕ0 = sin ϕ cos ϑ sin θ sin φ + cos φ sin θ sin φ + sin φ sin φ cos θ (5.16)

cos ϑ0 = − sin ϑ sin θ cos φ + cos ϑ cos θ, (5.17)

or to express (θ, φ) in terms of (ϑ0, ϕ0)

sin θ cos φ = cos ϑ cos ϕ sin ϑ0 cos ϕ0 + cos ϑ sin ϕ sin ϑ0 sin ϕ0 − sin ϑ cos ϑ(5.18)0

sin θ sin φ = − sin ϕ sin ϑ0 cos ϕ0 + cos ϕ sin ϑ0 sin ϕ0 (5.19)

cos θ = sin ϑ cos ϕ sin ϑ0 cos ϕ0 + sin ϑ sin ϕ sin ϑ0 sin ϕ0 + cos ϑ cos ϑ0(5.20)

In the special cases of (1) the observer is on the polar axis (ϑ = 0) and (2) the observer in the equatorial plane (ϑ = π/2 & ϕ = 0) we obtain closed form of GW strain (Eq. (5.7)) but in general the relation above should be used in an implicit form.

59 5.3 DIRECTION DEPENDENT NEUTRINO LUMINOSITY DL/DΩ

In axisymmetry, Kotake et al. [9] uses the following formula for evaluation of the dL/dΩ.

Z Rs dL 2 = drr |Qν|, (5.21) dΩ 0

where Rs is the outermost radius of the computational domain. The integration is

performed in each angular direction. Qν is the total neutrino cooling. In the integral (5.21), neutrinos are assumed to be emitted radially in each angular bin. Since this is a very restrictive assumption, we choose an alternative approach to estimate dL/dΩ factor. ~ Let Fν(~x,t, Eν) be the ﬂux of the corresponding type of neutrinos at the outermost

points (~x) of computational domain and Eν their energy. The total luminosity of neutrino emission is then

X Z Z ~x L(t) = dE · F~ dS, (5.22) ν |~x| ν ν S or Z " Z # 2 X ~ L(t) = dΩ Rs dEνFν · ~n, (5.23) Ω ν where ~n = ~x/|~x| is an outward pointing unit vector. From Eq. (5.23) we get

dL X Z ≡ R2 dE (F~ · ~n). (5.24) dΩ s ν ν ν In this approach, dL/dΩ(ϑ0, ϕ0, t0) is calculated using just the neutrino ﬂux computed at the outermost zone of computational grid.

60 5.4 NUMERICAL IMPLEMENTATION

5.4.1 Computation of N20

In order to compute N20, our GW code performs the integration of Eq. (5.6). CHIMERA data from 2D simulations, which are unequally-spaced in the radial direction and equally-spaced in the θ-direction, are either (1) spline-interpolated and integrated by spline-quadrature in θ, or (2) spline-interpolated (on Gauss-Legendre collocation points) and integrated by Gauss-Legendre quadrature. Then, they are integrated in the radial direction. The result is multiplied by the corresponding tensor

TT TT harmonics f20 to get h+ and h× .

5.5 VERIFICATION TESTS

We have tested the GW extraction subroutines with two tests. These are two artiﬁcial problems that can be solved analytically (see Appendix C).

5.5.1 Oscillating ring around a central mass

For the ﬁrst test, we consider a thin uniform ring orbiting around a central mass (Figure 5.3). Neglecting the self-gravity of the ring, we focus on its radial oscillations. The oscillation frequency is the epicyclic frequency of the orbit, which is equal to the Keplerian frequency in Newtonian theory.

We can easily obtain the analytic form of A20 – the only non-zero component in

TT the spectral decomposition of hij in axisymmetrical case (Appendix C).

16Gr π m A = R2Ω2α(cos Ω t + α cos 2Ω t), (5.25) 20 c4 15 2π 0 0 0 0

where m is mass of the ring, R0 its radius, Ω0 the Keplerian frequency, α the linear amplitude of oscillation. We evolve the system with this oscillation frequency and

61 Figure 5.3: Oscillating ring test.

compute A20. This test serves to check the terms related to the radial component of the velocity. The numerical results are in good agreement with analytical solution (Figure 5.4). However, we have to make a few remarks on the results. To model numerical data for density and velocity we assumed a small (compare to radius of ring) but finite thickness of the ring. The analytical expression above is valid only for infinitesimally thin rings, so the numerical data is already an approximation to the true solution. Equation (5.25) was derived under assumption that density of the ring is constant. Since the thickness of the ring changes due to its oscillations we change the cross- section of the ring artificially to keep its density constant. We have also a discontinuity of the density at the surface of the ring. Since we perform simulations on a finite grid, the jump produces numerical noise in the A20-function computed by the code. Finally, the ring is localized near the equatorial plane. This setup tests the worst- case scenario for our GW code, which is designed for a spherical grid. Low θ-resolution would produce significant errors. However, in the case of supernova scenarios, we deal with configurations where mass is distributed more or less uniformly in the θ-direction,

62 Figure 5.4: A20 amplitude of gravitational wave caused by ring oscillations. The central mass is M = 10M , the ring mass is m = 1M , the equilibrium radius of ring is R0 = 6.0Rg, and the thickness of the ring is 0.1R0. The amplitude is α = 0.2. The grid points are nx = 512 in radial direction, and ny = 512 in angular direction (0 ≤ ϑ0 ≤ π).

63 Figure 5.5: Ellipsoid rotating around z-axis.

which is more favorable for our code.

5.5.2 Rotating triaxial ellipsoid

The second test is the calculation of GW signals produced by the rotation of a triaxial ellipsoid with angular frequency Ω around the z-axis (Figure 5.5). This test complements the ﬁrst test checking the toroidal component of the velocity. We set up a triaxial ellipsoid whose surface is given by

x2 y2 z2 R + R + R = 1, (5.26) a2 b2 c2

where (xR, yR, zR) are Cartesian coordinates in a co-rotating frame. For the density of the ellipsoid, we use the following function

r x2 y2 z2 ρ = ρ F (r); ρ = const., r = R + R + R , (5.27) 0 0 a2 b2 c2 and choose the following density profile2 sin πr F (r) = . (5.28) πr 2 We first worked on the constant density case, which has for solution a Jacobi ellipsoid. This model has a density discontinuity at its surface, which create a high-frequency noise in the signal computed by a finite grid.

64 Figure 5.6: rh+ and rh× signal from rotating triaxial ellipsoid. The central density is 1014g/cm3, three axes are a = 3 × 106(cm), b = 2 × 106(cm), c = 106(cm). Rotational frequency is Ω = 0.5(/sec). The observer’s position angle is ϑ = π/6.

The analytic form of the signal is calculated in Appendix C. Assuming that the observer is at ϕ = 0 and ϑ = 0 in S-frame, we ﬁnd the gravitational strains for both polarizations:

Gρ 8 6 rh = 0 · 1 − abc(b2 − a2)Ω2 cos 2Ωt · (1 + cos2 ϑ) (5.29) + c4 3 π2 Gρ 16 6 rh = 0 · 1 − abc(b2 − a2)Ω2 sin 2Ωt · cos ϑ (5.30) × c4 3 π2

In ﬁgure 5.6, we plot the rh+ and rh× components of the GW signals and compare them with an analytic solution.

65 CHAPTER 6

RESULTS

The results presented in this chapter have been summarized and published in Gravi- tational Waves from Core Collapse Supernovae by Konstantin Yakunin, Pedro Mar- ronetti, Anthony Mezzacappa, Stephen Bruenn, Ching-Tsai Lee, Merek Chertkow, Raphael Hix, John Blondin, Eric Lentz, Bronson Messer, and Shin Yoshida, Class. Quant. Grav., 27: 194005, (2010) [10].

6.1 GRAVITATIONAL WAVE EMISSION BY ASPHERICAL MASS MOTIONS

We performed two-dimensional simulations beginning with 12, 15, and 25M non- rotating progenitors [104] with axisymmetry imposed and resolutions of 256 (adaptive) radial and 256 angular zones. In addition, in the 15M progenitor case simulations were performed with resolutions of 512 (adaptive) x 128 and 512 (adaptive) x 256 zones. The code conserves total energy (gravitational, internal, kinetic, and neutrino) to within ±0.5 × 1051, given a conservative gravitational potential. CHIMERA currently has three main components: a hydro component, a neutrino transport component, and a nuclear reaction network component. The hydrodynamics is evolved via a Godunov finite-volume scheme, specifically, a Lagrangian remap implementation of the Piecewise Parabolic Method (PPM) [74]. A moving radial grid option wherein the radial grid follows the average radial motion of the fluid makes it possible for the core infall phase to be followed with good resolution. Following bounce, an adaptive mesh redistribution algorithm keeps the radial grid between the core center

66 and the shock structured so as to maintain approximately constant ∆ρ/ρ. For 256 and 512 radial zones, this ensures that there are at least 15 and 30 radial zones per decade in density, respectively. Gravity is computed by an approximate general relativistic potential for spherical gravity and Newtonian 2D/3D spectral Poisson solver for the higher moments. The equation of state (EOS) of Lattimer and Swesty [93] is currently employed for matter in the nuclear statistical equilibrium (NSE) above 1.7 × 108 g/cm3. Below this density matter in NSE is described similarly by 4 nuclei (neutrons, protons, helium, and a representative heavy nucleus) in a highly modified version of the EOS described by [94]. For regions not in NSE, an EOS with a nuclear component consisting of 14 alpha particle nuclei (4He to 60Zn), protons, neutrons, and an iron-like nucleus is used. An electronpositron EOS with arbitrary degeneracy and degree of relativity spans the entire densitytemperature regime of interest. The code implements neutrino transport using “ray-by-ray-plus”approximation [85], whereby the lateral effects of neutrinos such as lateral pressure gradients (in optically thick conditions), neutrino advection, and velocity corrections are taken into account, but transport is performed only in the radial direction. Transport is computed by means of multigroup flux-limited diffusion with a flux limiter that has been tuned to reproduce Boltzmann transport results to within a few percent [84]. All O(v/c) observer corrections have been included. The transport solver is fully implicit and

solves for four neutrino ﬂavors simultaneously (i.e., νes,ν ¯es, νµs and ντ s (collectively

νx s), andν ¯µs andµ ¯νs (collectivelyµ ¯xs)), allowing for neutrinoneutrino scattering and pair-exchange, and diﬀerent ν andν ¯ opacities. The PPM technology has been directly applied to both the spatial and energy advection of neutrinos in both the radial and lateral directions. The neutrino opacities employed for the simulations are the “standard”ones described in [36], with the isoenergetic scattering of nucleons replaced by the more exact formalism of [86], which includes nucleon blocking, recoil,

67 and relativistic eﬀects, and with the addition of nucleonnucleon bremsstrahlung [87] with the kernel reduced by a factor of ﬁve in accordance with the results of [88]. The nuclear composition in the non-NSE regions of these models is evolved by the thermonuclear reaction network of [89]. This is a fully implicit general purpose reaction network; however, in these models only reactions linking the 14 alpha nuclei from 4He to 60Zn are used. Data for these reactions is drawn from the REACLIB compilations [92]. The nucleons have only very small abundances at any time and are included to make the NSE – non-NSE transition smoother. The iron-like nucleus is included to conserve charge in a freeze out occurring with an electron fraction below 0.5 [cf. 20]. The advection of material across an NSE – non-NSE interface in either direction is performed as detailed in [72]. Also, entire zones are moved from NSE to non-NSE as conditions dictate. Successful explosions are obtained in all simulations, with the longest running

51 model (the 25M model) having an explosion energy of 0.7 × 10 ergs (and still growing) 1.2 seconds after bounce. While the GW emissions we predict diﬀer in detail from model to model, a clear GW signature composes of four parts (Figure 6.1).

6.1.1 Prompt signal

In all our models collapse proceeds in a spherically symmetric way and, hence, it does not produce any GW signal. Just after core bounce the outgoing shock wave leaves behind negative entropy and lepton gradients. The region behind the shock is Ledoux unstable and develops prompt post-shock convection, which leads to an

early burst of gravitational wave signal with a peak amplitude of rh+ ∼ 5 cm lasting for 50 to 75 ms after bounce (left snapshot of Figure 6.2). During this time, the GW amplitude decreases to ∼ 1 cm, which is consistent with the results of Ott [22],

68 Figure 6.1: The left column shows the GW strain times the distance to the observer vs. post-bounce time for non-rotating progenitors of 12, 15, and 25 M . The signal is split into matter- (red-solid) and neutrino-generated (blue-dashed) signals. Note that the scales are diﬀerent for these two signals. The insets show the ﬁrst 70 ms after bounce. The right column shows the corresponding characteristic strain for both the matter (red) and the total (black) signals, compared to the AdvLIGO sensitivity curve. Figure from [10]

69 Figure 6.2: Top: Gravitational wave strain h+ times the distance to the observer r vs. post-bounce time for the 15M non-rotating progenitor model. Below, entropy distribution snapshots typical of the prompt, strong, and tail stages of the signal. Note the diﬀerence in scale of the left snapshot and two others. Figure taken from [11]

70 Figure 6.3: Contributions to the GW signal from two diﬀerent regions for the 15 M model: the PNS (r < 30 km) and the region above the PNS (r > 30km). The latter includes the region of neutrino-driven convection, the SASI, and the shock.

Marek et al. [13], and Murphy et al [12]. Basically, the prompt signal is generated by two independent phenomena: prompt convection inside the PNS generates a high- frequency signal that is superimposed on a lower-frequency component, seen in the insets of Figure 6.1. By using tracer particles, we ﬁnd that the latter is produced by the deﬂection of infalling matter while crossing the shock. This can be seen in

the inset in Figure 6.3, where the signal for our 15 M run has been split into the contributions from two diﬀerent regions, but it is in the tracer analysis of Figure 6.4 where this becomes evident. The matter-generated GW (solid red) is closely tracked by the GW generated by the infalling tracer particles deﬂected by the shock (dashed blue), some of which are shown in the left panel. The low-frequency signal from 20 ms to 60 ms after bounce originates at the shock radius, located at ∼ 100 km at this time and well outside the PNS. In the past, authors attributed the prompt signal to convection only [12,13].

71 Figure 6.4: Left: Trajectories of the tracer particles. It is shown the clear deﬂections of infalling particles through the shock that collectively produce low-frequency high- amplitude component of the GW signal shown on the right panel.Right: Comparison between the matter signal (solid red) and signal calculated using the tracers (dashed blue). Both panels correspond to our 15 M simulation.

6.1.2 Quiescent stage

When shock stalls near 180 km and the prompt convection ceased and prior to the development of neutrino-driven convection and the SASI, we observe an intermediate quiescent stage of GW signal with an amplitude of rh+ < 1 cm, which ends somewhere between 125 ms and 175 ms after bounce. Small values of the GW amplitude perclude the presents of violent convection.

6.1.3 Strong signal

A strong signal follows the quiescent stage and is the most energetic part of the GW signal. This phase starts at ∼ 150 ms after bounce and ends somewhere between 350 ms and 450 ms after bounce. This signal with the peak amplitudes of rh+ ∼ 30 cm, is dominated by SASI-induced funnels impinging on the PNS surface (top panel of Figure 6.5 and Figure 6.3). It shows evidence of two components: The low-frequency component arises from modulations in the shock radius as the SASI develops and

72 Figure 6.5: Top: Entropy distribution at 244 ms after bounce for the 15 M model. A large, low-entropy (blue-green) accretion funnel at an angle quasi-orthogonal to the symmetry axis and high-entropy (yellow-orange-red) outﬂows below the shock, along the symmetry axis, are evident. Bottom: Shock radius as a function of time for three regions: the north pole (solid blue), the equatorial plane (dotted black), and the south pole (dashed red).

73 Figure 6.6: Left: Trajectories of the tracer particles for 15 M simulation. The trajectories show how initially accreting matter is ejected by the revived shock and produces “tail”part of the GW signal. evolves, while the high-frequency component is generated when the SASI-induced accretion flows strike the PNS (see Figure 6.5)[12,13]. The shock modulations affect the kinetic energy of the accretion flows and, consequently, the amplitude of the GWs generated when these flows hit the PNS. Hence the high-frequency modulations are beneath a low-frequency envelope.

6.1.4 The “tail”part

The “tail”part of the GW is associated with global asymmetries in mass ejection (Figure 6.6) that start before the end of the strong signal at about 300 ms after bounce and last to the end of simulations. All of our GW signals end with a slowly increasing mean value of the GW amplitude, which reﬂects the gravitational memory associated with our prolate explosions. In turn, an oblate explosion would produce a negative slope for this tail. Geometry of the both types of explosions is presented on Figure 6.7. The tail continues to rise at the end of our runs because the explosions are still developing and strengthening. Focusing on the 15M model and Figure 6.3,

74 Figure 6.7: Snapshots of the entropy distribution during explosion for the three models representing each type of the explosion. The GW signal may indicate that the explosion is “spherical”, “oblate”, or “spherical”. Figure taken from [12].

75 the explosion initiates at ∼ 300−350 ms after bounce, and by 400 ms the signal from the PNS has largely ceased. Thus, the total GW signal begins to rise at ∼ 300 ms after bounce still a superposition of contributions from both above and below 30 km and eventually originating only from the region above 30 km. Figure 6.3 shows that convective instabilities inside the PNS make the main contribution into the GW amplitude for the ﬁrst three phases of the GW signal, and the tail is generated by the matter swept by the expanding shock during the explosion. Waveforms covering the ﬁrst three phases have been computed by Marek et al. (Garching group) [13]. They have performed their 2D simulations with the

Prometheus-Vertex code for a non-rotating 15M progenitor [105]. The hydrodynamics module of this code is based on a conservative and explicit Eulerian implementation of a higher order Godunov-type scheme. General relativistic gravity is approximated by the effective GR potential [78]. The neutrino transport is computed with the same “ray-by-ray plus”scheme used in CHIMERA [85]. They have modeled the GW signal using two sets of EOS: asoft version of the Lattimer & Swesty EOS and the considerably stiffer Hillebrandt & Wolff (HW) EOS [106]. They have found that p the prompt part of the GW signal has peak amplitudes of rh+ = A20 15/64π ∼ 3 cm and a characteristic frequency around 70–100 Hz (Figure 6.8). They also found that the SASI sloshing of the shock and the associated quasi-periodic mass motions lead to a time modulation of the neutrino emission and to GW amplitudes whose size and characteristic frequency depend on the compactness of the PNS. They also observed dependence of the characteristic frequency peak on the compactness of the nascent neutron star and thus on the properties of the high-density EOS: the more compact neutron star, the more powerful shock oscillations and significantly larger GW amplitudes earlier after core bounce. Also the characteristic frequencies of the low-` SASI modes and of the GW signal are higher. The main peak of the wave

76 Figure 6.8: Gravitational wave quadrupole amplitudes A20 as functions of post-bounce time (top) and corresponding Fourier spectra (bottom) associated with mass motions for 15M model. Simulations are performed by Marek et al. [13]. Figure taken from [13].

77 spectrum is located at 300-600 Hz for the stiﬀer HW EOS, and at 600–800 Hz for the softer LS EOS.

Our GW predictions for the 15 M case can be compared to those of Marek et al. [13] given that both groups implement similar treatments of the neutrino transport (Figure 6.9), GR corrections to the gravitational field and include essentially the same overall multi-physics in their models. The two groups are in agreement with regard to the time scales of the different (pre-explosion) GW phases, the amplitude of the prompt signal, and the peak in the GW spectrum at ∼ 700 − 800 Hz for LS EOS. They differ, however, in their predictions for the amplitude of the strong signal, where our results are about twice as large, and in the absence of the tail since no explosions were produced in the simulations of the Garching group. These differences likely arise in part due to the different progenitors used (Woosley & Weaver 2007 in CHIMERA vs. Woosley & Weaver 1995 in Marek et al.), which in turn alters the time scale of the explosion and, consequently, the amplitude of the GW signal in the strong-signal phase at any instant of time. Waveforms covering all four phases and based on parameterized explosions were reported in Murphy et al. [12]. The parameterized explosions are artificially initi- ated by means of a proper adjustment of parameters used in their simplified neutrino scheme. Instead of solving the Boltzmann transport equation, this scheme approx- imates neutrino heating and cooling by local quantities and predefined parameters such as density, temperature, the distance from the center, the electron-neutrino temperature, etc. However, their neutrino luminosities are kept isotropic and constant throughout their simulations [12]. The overall shape of the GW signatures from Murphy et al. (Figure 6.10) reflect the stages shown in Figure 6.1. A comparative analysis of Murphy et al. is more difficult given that their models are parameterized and can only be made on a qualitative level. In order to compare

78 Figure 6.9: Luminosities of electron neutrinos versus time as measurable for a distant observer located along the polar axis of the 2D spherical coordinate grid. Comparison between 15M simulation of Garching group [13](top) and CHIMERA simulation (bottom). The top panel of Figure taken from [13].

79 Figure 6.10: A sample of GW strain (h+ ) times the distance, D, vs. time after bounce. This signal was extracted from a simulation of Murphy et al. using a 15 M progenitor 52 model (Woosley & Heger 2007) and an electron-type neutrino luminosity of Lνe = 3.7×10 erg/s. Figure taken from [12].

80 Figure 6.11: Trajectories of the tracer particles for 15 M simulation. The color lines track trajectories of the particles that correspond to the downflow plume hitting the PNS surface and bouncing back. the GW signals we use the phenomenological model developed by Murphy et al. [12]. We argue that the funnels impinging on the PNS surface correlate in time with the spikes in the GW signature. Hence, one can conclude that the strength and characteristic frequencies of the GW signals are predominantly defined by the dynamics of the funnels. Murphy et al. have analyzed the downdraft motion based on theory of convection and buoyancy [12]. They note that buoyancy forces at the boundary between convective and stably-stratified regions determine the deceleration of the funnel and, hence, the GW amplitudes and frequencies. Following this, we define the PNS “surface”as the region where the negatively buoyant downdrafts become positively buoyant and reverse their downward motions. This is illustrated by the trajectories of the tracer particles on Figure 6.11. Murphy et al. find that the GW strain produced by the funnel can be roughly estimated by 4πG h ∼ ρf v r3∆r sin θ∆θ, (6.1) + Dc4 p p where vp is the typical velocity of the funnel and fp is its characteristic frequency. This characteristic frequency fp is inversely proportional to the dynamical timescale (Eq.

81 3.20) and, therefore, directly proportional to the square root of the average density (compactness of the PNS) which is strongly dependent upon both the gravitational potential and the dense-matter EOS [13]. We use the eﬀective GR potential and the soft Lattimer & Swesty EOS in our simulations while Murphy et al. use the Newtonian potential and the stiﬀ Shen EOS in their simulations. As a result, the compactness of the PNS as well as accretion velocities are larger in our models. In light of this phenomenological model, one can expect GW amplitudes (Eq. 6.1) and frequency peaks higher than those from our simulations (compare the left panel on Figure 6.1 and Figure 6.12). The total emitted GW energy is shown in Figure 6.13. For the more massive progenitors, all of the GW energy is emitted between 200 ms and 400 ms after bounce.

For the 12 M case, the GW energy is emitted more slowly, consistent with the fact that the explosion in this case unfolds more slowly [71]. Our predictions are 20 to 50 times larger than those of Murphy et al., but this is consistent with our waveforms having two to three times the amplitude and higher frequencies than the signals they predict (Eq. 3.25). The work presented here takes natural next step beyond this earlier, foundational work. A more precise prediction of the GW amplitudes and the timescales associated with each of the four phases requires a non-parameterized approach. Even in the case of a non-parameterized approach, prior to evidence of an explosion it is diﬃcult to assess whether or not the amplitudes and timescales are well determined. Thus, the non-parameterized explosion models studied here enable us to predict all four phases of the GW emission and their amplitudes and timescales with some conﬁdence.

Considering hchar (right column of Figure 6.1), it is important to note that the

peak at ∼ 700 − 800 Hz is associated with the high-frequency component of h+, which in turn is associated with the SASI-induced downﬂows hitting the PNS surface

82 Figure 6.12: hchar (Eq. 3.24) vs. frequency for the suite of simulations presented in Murphy et al. The spectra show broad peaks and some dependence upon the progenitor mass: ∼300 Hz for 12 M and ∼400 Hz for 40 M . For comparison, the approximate noise thresholds for Initial LIGO (solid-black curve), enhanced LIGO (dot-dashed-black curve) and Advanced LIGO (dashed-black curve) are plotted. Figure taken from [12].

83 Figure 6.13: Energy emitted by GWs during the ﬁrst 500 ms after bounce for all three models presented here. as discussed above. A precise association of the signal at lower frequencies with phenomena in the post-bounce dynamics will require a detailed analysis using tracer particles and is work in progress. The lower-frequency modulations (the envelope) of h+, which in turn are associated with the SASI-induced shock modulations will certainly be an important component of this lower frequency signal.

6.2 GRAVITATIONAL WAVE EMISSION BY ANISOTROPIC NEUTRINO RA- DIATION

As it has been shown in section 5.2, anisotropic neutrino emissions may also produce gravitational waves, which are significantly different from signals by mass motions. In Figure 6.1 (left panel), we present GW waveforms due to anisotropic neutrino radiation (blue dash line) for our non-rotating progenitors. For all three models we found common properties for the neutrino GW signals. Owing to a symmetric distribution of the neutrino luminosity (top panel of Figure 6.14) these signals are almost negligible compared to the matter ones up to ∼150 ms after bounce (Figure 6.1) when a relatively stable accretion downflow disk located at 60◦ < ϑ < 120◦ forms (top panel of Figure 6.5). The downflow, which carries cold,

84 Figure 6.14: Snapshots of the total neutrino luminosity for the 15 M model. At the top panel you can see almost spherical symmetric distribution of the neutrino luminosity at 125 ms after bounce. The middle panel shows the clear asymmetry in the neutrino luminosity due to the accretion downﬂow that increases neutrino opacities at the equatorial zone at 381 ms after bounce. The bottom panel illustrates the distribution of the neutrino luminosity at 533 after bounce that leads to decreasing in the amplitude of the GW signal.

85 dense matter within the central zone of the core, splits the whole domain into three zones: north, south and equatorial. For this matter distribution, the dynamical effect of a spherically-symmetric neutrino radiation field is enhanced along the low-density polar directions (middle panel of Figure 6.14). As a result of the risen asymmetry and the inherent memory effect (Eq. 5.7), the amplitude of the GW signal from neutrinos grows with time dramatically. The sign of the amplitude is defined by the function Ψ(ϑ) in Eq. (5.10). In the described configuration, neutrinos mainly radiate in directions where the function Ψ(ϑ) > 0 (Figure 5.2). Therefore, the resulting DC offset of the GW signal amplitude is positive. The behavior of the A20(t) function changes at about 450 ms after bounce because the neutrino luminosity tends to be again more spherically symmetric at the end of the simulation (bottom panel of Figure 6.14). Mülleret al. [14] have modeled the emission of GW from anisotropic neutrino radiation for rotating progenitors. The signal is shown in Figure 6.15. In spite of the difference in time evolution, one can observe a qualitative agreement between signals. In the simulations of Mülleret al., the asymmetric collapse produces the GW burst just after bounce. For GW from mass motions, rotation strongly increases the signal at bounce and during the convective phase. However, the neutrino signal does not increase as dramatically. Kotake et al. [16] studied for the first time the properties of GW of 3D parameterized models. They used the ZEUS-MP code [107] to describe the dynamics of the standing accretion shock flows of matter around the PNS. Neutrino emission has been modeled by means of the light-bulb approximation [39], where the neutrino luminosities from the PNS were adjusted to trigger explosions. They showed that the waveforms vary more stochastically than for 2D because the explosion anisotropies depend sensitively on the growth of the SASI, which develops chaotically in all di-

86 E2 Figure 6.15: Gravitational wave quadrupole amplitude A20 by M¨ullleret al. [14] vs. time (post-bounce) due to convective mass ﬂow and anisotropic neutrino emission (thin line) for the rotating delayed explosion model s15r of Buras et al. (2003) [15]. The inset shows an enlargement of the signal around the time of bounce. Figure taken from [14].

87 Figure 6.16: Gravitational wave signals are produced in the simulations of Kotake et al. [16] The top panel shows the GW spectrum contributed from neutrinos (solid) and from the matter (dashed) in a rotating model with Ω = 4 rad−1 imposed initially on a 15M progenitor model. In the bottom panel, the open circles and the pluses represent the amplitudes of hν, eq with the characteristic frequencies of νeq for the models with the cylindrical and the shell-type rotation proﬁles, respectively. Under the frequency of νeq, the GWs from the neutrinos dominate over those from the matter contributions. From the panel, it is seen that the GWs from neutrinos dominate over the ones from the matter in a lower frequency (f ≤ 100 Hz). Note that the source is assumed to be located at the distance of 10 kpc. Figure from [16]. rections. They also observed that the signal from CCSN is dominated at nearly all frequencies by the matter contributions (Figure 6.16). However, during the SASI phase, Kotake et al. found that the amplitude of the GW signal from neutrino emission is about twice as much as those from convective matter motions (Figure 6.17).

In spite of the presence of the axisymmetry, the results of Kotake et al. have been conﬁrmed by our CHIMERA 2D simulations. From the spectrum analysis (right panel of Figure 6.1), one can see that the neutrino GWs, though dominant over the

88 Figure 6.17: Gravitational waveforms produced by the 3D simulations of Kotake et al. [17] from neutrinos (bottom) and from the sum of neutrinos and matter motions (top), seen from the polar axis and along the equator (indicated by “Pole”and “Equator”) with polarization (+ or × modes). The distance to the SN is assumed to be 10 kpc. Figure from [17]. matter GWs in the lower frequencies below 20 Hz, become very diﬃcult to detect for ground-based detectors whose sensitivity is limited mainly by the seismic noises at such lower frequencies [108]. We conclude that GW signals from anisotropic neutrino emission has several features, which allows us to diﬀerentiate them from GW matter signals. Neutrino waveforms dominate the amplitudes after 150 ms. and have less variation in the time structure, producing a stronger signals at lower frequencies.

89 CHAPTER 7

CONCLUSIONS AND FUTURE WORK

We presented gravitational waveforms computed in the context of 2D core collapse supernova simulations performed with the CHIMERA code for non-rotating 12, 15, and 25M progenitors. We calculated the contribution to the signals produced by both baryonic matter motion and anisotropic neutrino emission up to 530 ms after bounce for all three progenitors. Given the development of non-parameterized explosions in our models, we are able to compute the waveforms through explosion and to determine more precisely the pre-explosion amplitudes and timescales. Given our use of tracer particles, we are able to decompose the GW signatures and determine which phenomena contribute to specific components of the waveforms. This allowed us to identify an additional source for the prompt signal (in the past solely attributed to prompt convection) the deflection of infalling matter through the shock. Our waveforms exhibit a characteristic signature. Namely, the signal develops in four stages. There is 1) a relatively short and weak prompt signal, 2) a quiescent stage, 3) a strong signal where most of the GW energy is emitted and, 4) a slowly increasing tail. We predict signatures with sufficient strength to be readily observable by Advanced LIGO for a Galactic event, and the peak in the observable spectrum stems from the accretion downflows driven by the SASI. The results presented here are preliminary: a new set of 2D simulations performed with an enhanced version of our CHIMERA code is currently ongoing [71]. The 2D version of CHIMERA is currently most advanced existing CCSN codes,

90 does not yet include all possible ingredients needed to a full description of the supernova explosion. What is missing? We list the components that have to be added into the future CCSN codes in decreasing order of importance in opinion of the leading researchers in area of CCSN simulations. [18–20,109].

• 3D models

Three-dimensional models are required. We anticipate that the greatest change to our gravitational waveform predictions in moving to 3D will be in the phase- 4 tail. Prolate explosions are often seen in axisymmetric simulations, where artificial boundary conditions must be imposed that prevent the turnover of material along the symmetry axis. With axisymmetry removed, we expect a significant change in the evolution of the explosion tail: in its magnitude, and perhaps even in its sign. And no doubt there will be quantitative changes to the amplitudes and timescales associated with earlier phases, particularly in the strong signal arising from the SASI motions. In 3D, the SASI is dominated by spiraling flows [46], fundamentally different than the sloshing modes that dominates in the axisymmetric case. This will in turn alter the waveforms in the final pre-explosion phase. This has already been demonstrated by Kotake et al. in 3D parameterized studies [17].

In a very recent, paper M¨ulleret al. [110] have presented results of neutrino- driven supernova explosions in the 3D parametrized models of non-rotating

15M and 20M progenitors. They have found that 3D wave amplitudes are smaller by a factor of 23 than those predicted by 2D models [10, 12–14] due to less coherent mass motions and neutrino emission. They also observed strong dependence between the polarizations, diﬀerent explosion simulations, diﬀerent

91 observer directions and the corresponding waveforms. In spite of the simpliﬁed treatment of neutrino transport and the excision of the high-density core of the PNS in these 3D models, the results of M¨ulleret al. emphasize the importance of the 3D simulations of CCSN.

The 3D simulations with CHIMERA including the most important physics doc- umented above are ongoing, and we look forward to reporting on their GW signatures in the near future.

• Improved Neutrino Transport

Neutrino transport is one of the most important components of any supernova model. Neutrino transport is generally described by the neutrino Boltzmann kinetic equations. The accurate treatment of neutrino physics is essential for the neutrino-heating mechanism, where neutrinos emitted from the hot dense PNS re-energize the stalled shock and induce the explosion [111]. Because the heating rate depends on the energy and angle distributions of the neutrinos (Eq. 2.6), neutrino radiation transport is a seven-dimensional problem: six dimensions of phase space plus the neutrino energy spectrum. This is what makes core-collapse supernova a major computational challenge requiring exascale supercomputers. Current state of the art neutrino transport in supernova simulations [3, 13] is done with “ray-by-ray-plus”approximation, where transport is performed only in the radial direction. This approximation has to be improved in future more realistic CCSN modeling.

• full GR

All current multi-dimensional CCSN simulations use either the Newtonian potential or the approximate relativistic potential. We already discussed the important inﬂuence of GR eﬀects on the dynamics of supernova explosion and,

92 as a consequence, on the generated GW waveforms. The effective potential that we use in the CHIMERA simulations produces the first-order GR effects. However, the completely consistent CCSN model has to include an numerical algorithm, such as the BSSNOK [112–114], to evolve hydrodynamics in full GR framework.

• Rotation

Most of the previous calculations of gravitational waves have focused on the bounce signals for rotational progenitors [115–120]. Due to the inherent asphericity of the rotational progenitors, the prospective GW burst signal is already produced at an early postbounce phase (< 30 ms after bounce). However, the current simulations with the rotational progenitors use unrealistically high rotation [121, 122]. Nevertheless, rotation has to be considered because even in the case of the slowly rotating supernova cores, initial asphericity produces stronger signals at bounce compared to the one from non-rotating cores. Grav- itational waveforms with few clear peaks (bounce and SASI) suit much better for the GW burst search algorithms. The ﬁrst peak in the signal acts as a trigger to look for the next one within the well restricted time interval (∼ 230 ms from bounce to the SASI). It signiﬁcally increases sensitivity of the searching algorithm, and as the result, it makes this type of signals more likely to be detected [123].

• Magnetic ﬁelds

While the importance of magnetic fields to the supernova explosion mechanism remains unclear, combined effects of flux-freezing, field winding and the magneto-rotational instability [124] lead to the growth of magnetic pressure that can help in the re-energizing of the stalled shock. The magnetic fields could also

93 convert the free energy of the diﬀerential rotation of the forming compact remnant to kinetic energy of the SN ejecta. The magneto-rotational mechanism may also be relevant in the context of long-soft gamma ray bursts [125]. Taking all these into account one can conclude that the magnetohydrodynamics will be required for any realistic CCSN code in near future.

94 APPENDIX A

TENSOR SPHERICAL HARMONICS FLM

In this appendix, we write explicitly the tensor spherical harmonics. We follow the notation of Kotake et al. [9].

A.1 EXPLICIT FORM OF WLM AND XLM

In general, the Wlm and Xlm functions are deﬁned by

∂2 ∂ 1 ∂2 W (θ, φ) = − cot θ − Y (θ, φ) (A.1) lm ∂θ2 ∂θ sin2 θ ∂φ2 lm ∂ ∂ X (θ, φ) = 2 − cot θ Y (θ, φ), (A.2) lm ∂ϕ ∂θ lm

where Ylm is the usual scalar spherical harmonics of order l and degree m. For l = 2 (quadrupole approximation), we have the following explicit expressions

r 15 X = i sin θ cos θe2iφ (A.3) 2,2 2π r 15 X = i sin2 θeiφ (A.4) 2,1 2π

X2.0 = 0 (A.5) r 15 X = i sin2 θe−iφ (A.6) 2,−1 2π r 15 X = −i sin θ cos θe−2iφ (A.7) 2,−2 2π (A.8)

95 and r 15 1 + cos2 θ W = e2iφ (A.9) 2,2 2π 2 r 15 W = sin θ cos θeiφ (A.10) 2,1 2π r 15 W = 3 sin2 θ (A.11) 2,0 4π r 15 W = − sin θ cos θe−iφ (A.12) 2,−1 2π r 15 1 + cos2 θ W = e−2iφ (A.13) 2,−2 2π 2 (A.14)

A.2 TENSOR SPHERICAL HARMONICS

Let us define the tensor spherical harmonics (in the observer’s frame) using the Wlm and Xlm functions.   Wlm Xlm flm = α   . (A.15) 2 Xlm −Wlm sin θ Here the first row (column) corresponds to θ (φ). and α is a normalization factor. Notice that this tensor is trace-free and the diagonal component is the + mode while the off-diagonal components corresponds to the × mode. The normalization is fixed by the following relation

Z ∗ 2 AC 2 BD dΩ(flm)AB (fl0m0 )CD ( γ) ( γ) = δll0 δmm0 , (A.16)

where A, B, C, D = θ, φ and metric on the 2-sphere

2 γAB is   1 0 2 γAB =   . (A.17) 0 sin2 θ

The normalization factor for l = 2 is √1 . 4 3

96 APPENDIX B

TRANSVERSE TRACELESS PART OF “DIRECTIONAL”TENSOR

In order to take the transverse traceless part of the tensor product, we use the projection tensor Pab (Eq. 3.12). In our case, the projection is done on the plane perpendicular to the direction to the source (z-direction of the O-frame).

b b b Pa = δa − zˆazˆ , (B.1)

wherez ˆ is the unit vector along z-axis of the O-frame. One can deﬁne the projected tensor product

i j νab ≡ ninjPaPb , (B.2)

TT which is a tensor on the hypersurface perpendicular toz ˆ. Then, (ninj) can be

expressed in terms of the tensor νij 1 (n n )TT = ν − P · tr(ν). (B.3) i j ij 2 ij Here tr(ν) is computed as

k lr 2 tr(ν) = nknlPr P = 1 − nznz = 1 − cos θ, (B.4)

where θ is the direction angle of the neutrino emission measured in the O-frame. Note that this θ is the same angle that appears in the denominator of the integrand in Eq.(5.7).

TT Finally, (nxny) is given by 1 (n n )TT = n P in P j − P tr(ν) x y i x j y 2 xy 1 = (1 − cos2 θ) sin 2φ. (B.5) 2

97 Then, the “cross”-component of the GW strain can be written

Z t−R/c Z TT TT 2G 0 dL 0 0 0 h× ≡ hxy = 2 dt dΩ (1 + cos θ) sin 2φ (ϑ , ϕ , t ). (B.6) c R −∞ dΩ

Similarly we ﬁnd the “plus”-component

1 (n n )TT = (1 − cos2 θ) cos 2φ. (B.7) x x 2

Z t−R/c Z TT TT 2G 0 dL 0 0 0 h+ ≡ hxx = 2 dt dΩ (1 + cos θ) cos 2φ (ϑ , ϕ , t ). (B.8) c R −∞ dΩ

Note that θ and φ are implicitly related to ϑ0 and ϕ0 through the observer’s direction angles (ϑ, ϕ).

98 APPENDIX C

ANALYTICAL TESTS OF THE GW CODE.

C.1 ROTATING ELLIPSOID TEST

The following system of equations describes an ellipsoid in the spherical coordinates (Figure 5.5).   X = ar sin θ cos φ 0 ≤ r ≤ 1   Y = br sin θ sin φ 0 ≤ θ < π    Z = cr sin θ sin φ 0 ≤ φ < 2π R2 = X2 + Y 2 + Z2 = a2r2 sin2 θ cos2 φ + b2r2 sin2 θ sin2 φ + c2r2 cos2 θ (C.1)

The preferable density function for numerical modeling is one that has no discontinuities. We set the density of the ellipsoid in the following form:

sin πr ρ = ρ . (C.2) 0 πr

This choice allows us to avoid the discontinuity of the density function at the edge of the ellipsoid. In order to ﬁnd the gravitational signal from a rotating ellipsoid we consider the so-called reduced (TT) quadrupole moment

1 Z 1 ITT ≡ I − δ trI = dV¯ ρ X X − δ R2 (C.3) ij ij 3 ij i j 3 ij

∂ (X,Y,Z) dV¯ = drdθdφ = abc r2 sin θ dr dθ dφ (C.4) ∂(r, θ, φ)

99 Z Z 1 Z π Z 2π 2 ¯ 2 2 2 sin πr 2 2 2 Ixx = ρX dV = a abc ρ0 drr dθ sin θ dφ r sin θ cos φ 0 0 0 πr a3bcρ Z 1 Z 1 Z 2π = 0 dr r3 sin πr dµ 1 − µ2 dφ cos2 φ π 0 −1 0 | {z } | {z } =4/3 =π

Z 1 1 6 3 dr r sin πr = − 3 0 π π

From here we can ﬁnd all non-zero components of the quadrupole moment

4 1 6 I = − ρ a3bc (C.5) xx 3 π π3 0

4 1 6 I = − ρ ab3c (C.6) yy 3 π π3 0

4 1 6 I = − ρ abc3 (C.7) zz 3 π π3 0

4 1 6 tr(I) = − ρ abc (a2 + b2 + c2) (C.8) 3 π π3 0

4 1 6 1 4 1 6 I = − ρ a3bc − tr(I) = − ρ abc(2a2 − b2 − c2) (C.9) xx 3 π π3 0 3 3 π π3 0

4 1 6 I = − ρ abc(2b2 − c2 − a2) (C.10) yy 3 π π3 0

4 1 6 I = − ρ abc(2c2 − a2 − b2) (C.11) zz 3 π π3 0

Z Z 1 Z π Z 2π ¯ 2 2 2 2 Ixy = ρ XY dV = ab abc dr ρ r dθ sin θ dφ r sin θ sin φ cos φ = 0 0 0 0

100 Ixz = Iyz = 0

Let’s perform a coordinate transformation from the rotating frame (Eqs. C.1) to an inertial frame. Suppose the ellipsoid rotates around the z-axis with a constant angular velocity Ω. Therefore

      XI cos Ωt − sin Ωt 0 XR             Y  = sin Ωt cos Ωt 0 Y   I     R        ZI 0 0 1 ZR

I T R Iij = R ·Iij · R (C.12)

      cos Ωt sin Ωt 0 IR 0 0 cos Ωt − sin Ωt 0    xx    I       I = − sin Ωt cos Ωt 0  0 IR 0  sin Ωt cos Ωt 0 ij    yy       R    0 0 1 0 0 Izz 0 0 1

  IR cos2 Ωt + IR sin2 Ωt sin Ωt cos Ωt IR − IR 0  xx yy yy xx  I   I = sin Ωt cos Ωt IR − IR IR sin2 Ωt + IR cos2 Ωt 0  ij  yy xx xx yy   R  0 0 Izz

4 1 6 II = − ρ abc 2a2 − b2 − c2 cos2 Ωt + 2a2 − b2 − c2 sin2 Ωt (C.13) xx 3 π π3 0

  4 1 6 II =  −  ρ abc b2 − a2 sin Ωt cos Ωt (C.14) xy 3 π π3  0

101 4 1 6 II = − ρ abc 2a2 − b2 − c2 sin2 Ωt + 2a2 − b2 − c2 cos2 Ωt (C.15) yy 3 π π3 0

4 1 6 II = − ρ abc 2c2 − a2 − b2 (C.16) zz 3 π π3 0

4 1 6 I¨I = − ρ abc Ω2 − 2a2 − b2 − c2 2 cos 2Ωt + 2a2 − b2 − c2 2 sin 2Ωt xx 3 π π3 0 4 1 6 = − ρ abc Ω2 b2 − a2 cos 2Ωt (C.17) 3 π π3 0

I 8 1 6 I¨ = − ρ Ω2 a2 − b2 sin 2Ωt (C.18) xy 3 π π3 0

8 1 6 I¨I = − ρ abc Ω2 a2 − b2 cos 2Ωt (C.19) yy 3 π π3 0

¨I Izz = 0. (C.20)

Rotating the coordinate axes in the xz-plane to align the z-axis to the direction to the observer, we get

v = vxex + vzez (C.21)

0 0 0 v = vxex + vzez (C.22)

 0      vx cos ϑ − sin ϑ vx   =     0 vz sin ϑ cos ϑ vz

102 ¨I −1 ¨ Ii0j0 = R IR (C.23) i0j0

      ¨ ¨ cos ϑ 0 sin ϑ Ixx Ixy 0 cos ϑ 0 − sin ϑ       ¨I       I 0 0 =  0 1 0  I¨ I¨ 0  0 1 0  i j    yx yy          − sin ϑ 0 cos ϑ 0 0 0 sin ϑ 0 cos ϑ

  ¨ 2 ¨ ¨ Ixx cos ϑ Ixy cos ϑ −Ixx cos ϑ sin ϑ   ¨I   I 0 0 =  I¨ cos ϑ I¨ −I¨ sin ϑ  i j  xy yy xy    ¨ ¨ ¨ 2 −Ixx cos ϑ sin ϑ −Ixy sin ϑ −Ixx sin ϑ The projection operator is now given by

j0 j0 j0 Pi0 = δi0 − zˆi0 zˆ (C.24)

l0m0 ¨ ¨ 2 ¨ P Il0m0 = Ixx cos ϑ + Iyy (C.25)

¨TT We ﬁnally obtain expressions for Iij :

TT 2 1 2 1 2 1 I¨ 0 0 = I¨ cos ϑ − I¨ cos ϑ + I¨ = I¨ cos ϑ − I¨ (C.26) x x xx 2 xx yy 2 xx 2 yy

TT 2 1 2 TT I¨ 0 0 = I¨ cos ϑ − I¨ cos ϑ + I¨ = −I¨ 0 0 (C.27) y y yy 2 xx yy x x

¨TT ¨ Ix0y0 = Ixy cos ϑ (C.28)

1 1 1 I¨TT = I¨ cos2 ϑ − I¨ = 1 + cos2 ϑ I¨ (C.29) + 2 xx 2 yy 2 xx

103 ¨TT ¨ I× = Ixy cos ϑ (C.30)

4 6 I¨TT = 1 − ρ abc Ω2 b2 − a2 cos 2Ωt 1 + cos2 ϑ (C.31) + 3π π2 0

8 6 I¨TT = 1 − ρ abc Ω2 a2 − b2 sin 2Ωt cos ϑ, (C.32) × 3π π2 0 which allow for the calculation of the GW through the quadrupole formula (Eq. 5.1)

C.2 OSCILLATING RING TEST

In this section we derive an analytical expression of the gravitational wave strain produced by oscillations of an inﬁnitely thin ring of radius R surrounding a point mass M. GM 2 Let the ring have a linear mass density λ and an orbital frequency Ω0 = 3 . R0 The velocities of the ring points have only two components

rˆ φˆ v = v er + v eφ.

The Lagrangian of the ring is given by     I 1 GM 1 GM L = λdl  R˙ 2 + Rφ˙2 +  = m  R˙ 2 + Rφ˙2 +  . 2 R  2 R 

Since the Lagrangian is independent on φ, there is a constraint on the motion of the ring   d ∂L 2 ˙   = 0 ⇒ R φ = J0. (C.33) dt  ∂φ˙

104 The equation of ring motion is given by   d ∂L ∂L   − = 0. (C.34) dt ∂R˙  ∂R

GM R¨ − Rφ˙2 + = 0. (C.35) R2

2 J0 GM R¨ − + = 0. (C.36) R3 R2

2 4 2 Since J0 = R0Ω0 = GMR0 the last equation becomes

GMR0 GM R¨ − + = 0. (C.37) R3 R2

R 3 2 Let us introduce a new variable a = . Notice also that GM = R0Ω0. Then, R0

2 2 Ω0 Ω0 a¨ − + = 0 (C.38) a3 a2   1 1 a¨ + Ω2 − +  = 0 (C.39) 0  a3 a2

We will work in the ﬁrst order approximation, i.e. a = 1 + , where || 1.

2 ¨+ Ω0 (− (1 − 3) + (1 − 2)) = 0 (C.40)

2 −iΩ0t ¨+ Ω0 = 0 ⇒ = αe (C.41)

Here, α is the dimentionless parameter. Using the deﬁnition of the variable a, we get

R(t) = R0a = R0 (1 + α cos Ω0t) . (C.42)

rˆ ˙ v = R(t) = −R0α sin Ω0t. (C.43)

105 √ 2 ˙ R φ J0 GMR0 vφˆ = Rφ˙ = = = = R R R0 (1 + α cos Ω0t) v u uGM t = (1 − α cos Ω0t) . R0

m m λ = = (1 − α cos Ω0t) = λ0 (1 − α cos Ω0t) . 2πR 2πR0

In order to calculate N20, we deﬁne a density function of the ring

π m δ (r − R) δ θ − 2 ρ = . (C.44) 2πR r sin θ

Then, v u 32π2G Z π Z ∞ 1u 5 3 rˆ t 2 N20 = √ sin θdθ r dr ρv 3 cos θ − 1 = 4 5 3c 0 0 4 π v u 8π2Gu 1 mvrˆ Z π Z ∞ t 2 π 2 = 4 dθ r dr δ (r − R) δ θ − 3 cos θ − 1 = c 15π 2πR 0 0 2 v v u u 2 u 2 u 8π Gt 1 8π Gt 1 = − mvrˆR = − m (−R α sin Ω t) R (1 + α cos Ω t) = c4 15π c4 15π 0 0 0 0 v u 2 u 8π Gt 1 α2 = mR2Ω α sin Ω t + sin 2Ω t . c4 15π 0 0 0 2 0

Finally, we get an expression for the quadrupole component of the amplitude of the gravitational ﬁeld

v u 2 u dN20 8π Gt 1 A = = mR2Ω2α (cos Ω t + α cos 2Ω t) . (C.45) 20 dt c4 15π 0 0 0 0

106 BIBLIOGRAPHY

[1] A. Heger. How massive single stars end their life. Astrophysical Journal, 591:288–300, 2003.

[2] H.-Th. Janka, B. Mueller, F. S. Kitaura, and R. Buras. Dynamics of shock propagation and nucleosynthesis conditions in o-ne-mg core supernovae. Astro- physical Journal, 485:199–208, 2008.

[3] S. W. Bruenn, A. Mezzacappa, W. R. Hix, J. M. Blondin, P. Marronetti, O. E. B. Messer, C. J. Dirk, and S. Yoshida. Mechanisms of core-collapse supernovae & simulation results from the chimera code. AIP Conf. Proc., 1111:593– 601, February 2009.

[4] J. M. Blondin, A. Mezzacappa, and C. DeMarino. Stability of standing accretion shocks, with an eye toward core collapse supernovae. Astrophys.J., 584:971–980, 2003.

[5] L. Ju, D. G. Blair, and C. Zhao. Detection of gravitational waves. Rept.Prog.Phys., 63:1317–1427, 2000.

[6] http://www.ligo.caltech.edu/advligo.

[7] O. E. B. Messer, S. W. Bruenn, J. M. Blondin, W. R. Hix, A. Mezzacappa, and C. J. Dirk. Petascale supernova simulation with chimera. Journal of Physics: Conference Series, 68:012049, 2007.

[8] Ching-Tsai Lee. The Applications of the Tracer Particle Method to Multi- Dimensional Supernova Simulations. PhD thesis, The University of Tennessee, Knoxville TN USA, 2008.

[9] K. Kotake, N. Ohnishi, and S. Yamada. Gravitational radiation from standing accretion shock instability in core-collapse supernovae. Astrophysical Journal, 655:406–415, 2007.

[10] K. N. Yakunin, P. Marronetti, A. Mezzacappa, S. W. Bruenn, C.-T. Lee, M. A. Chertkow, W. R. Hix, J. M. Blondin, E. J. Lentz, O. E. B. Messer, and S. Yoshida. Gravitational waves from core collapse supernovae. Class.Quant.Grav., 27:194005, 2010.

107 [11] K. N. Yakunin, P. Marronetti, A. Mezzacappa, S. W. Bruenn, C.-T. Lee, M. A. Chertkow, W. R. Hix, J. M. Blondin, E. J. Lentz, O. E. B. Messer, and S. Yoshida. Core collapse supernovae using chimera: Gravitational radiation from non-rotating progenitors, November 2010. [12] J. W. Murphy, C. D. Ott, and A. Burrows. A model for gravitational wave emission from neutrino-driven core-collapse supernovae. Astrophysical Journal, 707:1173–1190, 2009. [13] A. Marek and H. Th. Janka. Delayed neutrino-driven supernova explosions aided by the standing accretion-shock instability. Astrophys.J., 694:664–696, 2009. [14] E. Mueller, M. Rampp, R. Buras, H.-Th. Janka, and D. Shoemaker. Towards gravitational wave signals from realistic core collapse supernova models. Astro- phys.J., 603:221–230, 2004. [15] R. Buras, M. Rampp, H.-Th. Janka, and K. Kifonidis. Improved models of stellar core collapse and still no explosions: What is missing? Phys.Rev.Lett., 90:241101, 2003. [16] K. Kotake, N. Ohnishi, S. Yamada, and K. Sato. Toward radiation- magnetohydrodynamic simulations in core-collapse supernovae. J.Phys.Conf.Ser., 31:95–98, 2006. [17] K. Kotake, W. Iwakami, N. Ohnishi, and S. Yamada. Ray-tracing analysis of anisotropic neutrino radiation for estimating gravitational waves in core-collapse supernovae. Astrophysical Journal, 704:951–963, 2009. [18] A. Burrows. Supernova explosions in the universe. Nature, 403:727, 2000. [19] A. Mezzacappa and G. Fuller, editors. Open Issues in Core Collapse Supernova Theory, volume 14. Institute of Nuclear Theory, Wold Scientiﬁc Publishing Co. Pte. Ltd., 2005. [20] K. Kotake, K. Sato, and K. Takahashi. Explosion mechanism, neutrino burst, and gravitational wave in core-collapse supernovae. Rept.Prog.Phys., 69:971– 1144, 2006. [21] H. A. Bethe. Supernova mechanisms. Reviews of Modern Physics, 62(4):801– 866, 1990. [22] C. D. Ott. The gravitational wave signature of core-collapse supernovae. Class.Quant.Grav., 26:063001, 2009. [23] Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler. Gravitation. San Francisco, W. H. Freeman, 1973.

108 [24] J. M. Weisberg and Taylor J. H. The relativistic binary pulsar b1913+16: Thirty years of observations and analysis. ASP Conf. Series, 328:25, 2005. [25] I. Leonor, L. Cadonati, E. Coccia, et al. Searching for prompt signatures of nearby core-collapse supernovae by a joint analysis of neutrino and gravitational-wave data. Class.Quant.Grav., 27:084019, 2010. [26] Massimo Turatto. Supernovae and Gamma-Ray Bursters. Springer, 2003. [27] W. D. Arnett, J. N. Bahcall, R. P. Kirshner, and S. E. Woosley. Supernova 1987a. Annual review of astronomy and astrophysics, 27:629–700, 1989. [28] R. A. Chevalier. The radio and x-ray emission from type ii supernovae. Astro- physical Journal, 259:302–310, 1982. [29] A. V. Filippenko. Optical spectra of supernovae. Annual Review of Astronomy and Astrophysics, 35:309–355, 1997. [30] Laurence A. Marschall. The supernova story. New York : Plenum Press, 1988. [31] B. Aschenbach. Discovery of a young nearby supernova remnant. Nature, 396:141, 1998. [32] S. van den Bergh and G. A. Tammann. Galactic and extragalactic supernova rates. Annual Review of Astronomy and Astrophysics, 29:363, 1991. [33] N. Arnaud, M. Barsuglia, M.-A. Bizouard, et al. Detection of a close supernova gravitational wave burst in a network of interferometers, neutrino and optical detectors. Astroparticle Physics, 21:201, 2004. [34] Stuart L. Shapiro and Saul A. Teukolsky. Black holes, white dwarfs, and neutron stars: The physics of compact objects. New York, Wiley-Interscience, 1983. [35] David Arnett. Supernovae and Nucleosynthesis: An Investigation of the History of Matter, from the Big Bang to the Present. Princeton Series in Astrophysics. 1996. [36] S. W. Bruenn. Stellar core collapse - numerical model and infall epoch. Astro- physical Journal Supplement Series, 58:771–841, 1985. [37] H. A. Bethe and Wilson J. R. Revival of a stalled supernova shock by neutrino heating. Astrophysical Journal, 295:14–23, 1985. [38] A. Burrows, J. C. Hayes, and B. Fryxell. On the nature of core collapse supernova explosions. Astrophys.J., 450:830, 1995. [39] H.-Th. Janka and E. Mueller. Neutrino heating, convection, and the mechanism of type-ii supernova explosions. Astron. Astrophys., 306:167, 1996.

109 [40] A. Mezzacappa, A. C. Calder, S. W. Bruenn, J. M. Blondin, M. W. Guidry, M. R. Strayer, and A. S. Umar. Investigation of neutrino-driven convection and the core collapse supernova mechanism using multigroup neutrino transport. Astrophys.J., 495:911–926, 1998.

[41] M. Liebendoerfer, A. Mezzacappa, F.-K. Thielemann, O. E. B. Messer, W. R. Hix, and S. W. Bruenn. Probing the gravitational well: No supernova explosion in spherical symmetry with general relativistic boltzmann neutrino transport. Phys.Rev. D, 63:103004, 2001.

[42] M. Liebendoerfer, O. E. B. Messer, A. Mezzacappa, W. R. Hix, and S. W. Bruenn. Probing the gravitational well: No supernova explosion in spherical symmetry with general relativistic boltzmann neutrino transport. Phys.Rev.D, 63:103004, 2001.

[43] M. Rampp and H.-Th. Janka. Spherically symmetric simulation with boltzmann neutrino transport of core collapse and post-bounce evolution of a 15 solar mass star. Astrophys.J., 539:L33–L36, 2000.

[44] M. Rampp and H.-Th. Janka. Radiation hydrodynamics with neutrinos: Vari- able eddington factor method for core-collapse supernova simulations. As- tron.Astrophys., 396:361, 2002.

[45] K. Kifonidis, T. Plewa, L. Scheck, H.-Th. Janka, and E. Mueller. Non-spherical core-collapse supernovae ii. late-time evolution of globally anisotropic neutrino- driven explosions and implications for sn 1987a. Astrophysical Journal, 453:661– 678, 2006.

[46] J. M. Blondin and A. Mezzacappa. Pulsar spins from an instability in the accretion shock of supernovae. Nature, 445:58–60, 2007.

[47] C. Papaliolios, M. Krasovska, L. Koechlin, P. Nisenson, and C. Standley. Asym- metry of the envelope of supernova 1987aasymmetry of the envelope of supernova 1987a. Nature, 338:565–566, 1989.

[48] P. Barrett and S. Ryan. Supernova 1987a in the large magellanic cloud. IAU Circ., 1:4337, 1987.

[49] M. Mendez, G. Clocchaiatti, C. Feinstein, and U.G. Marraco. Supernova 1987a in the large magellanic cloud. Astrophysical Journal, 334:295, 1988.

[50] H. E. Schwarz. Supernova 1987a in the large magellanic cloud. IAU Circ., 1:4338, 1987.

[51] Albert Einstein. Die grundlage der allgemeinen relativit¨atstheorie. Annalen der Physik, 49:769, 1916.

110 [52] Steven Weinberg. Gravitation and cosmology: principles and applications of the general theory of relativity. New York, Wiley, 1972.

[53] Robert M. Wald. General relativity. Chicago : University of Chicago Press, 1984.

[54] C. L. Fryer and K. New. Gravitational waves from gravitational collapse. Living Reviews in Relativity, 6, 2003.

[55] Sean Carroll. Spacetime and geometry an introduction to General Relativity. Addision Wesley: San Francisco, CA, 2004.

[56] Ignazio Ciufolini, Vittorio Gorini, Ugo Moschella, and Pietro Fre, editors. Grav- itational Waves. IOP Publishing Ltd, 2001.

[57] James Hartle. Gravity: an introduction to Einstein’s general relativity. Pearson Education: San Fransisco, CA, 2003.

[58] https://wwwcascina.virgo.infn.it.

[59] GEO600. http://www.geo600.uni-hannover.de/geocurves/, May 2011.

[60] http://tamago.mtk.nao.ac.jp.

[61] S. W. Hawking and W. Israel. 300 Years of Gravitation. Cambridge University Press, Cambridge, 1987.

[62] J. Hough and S. Rowan. Gravitational wave detection by interferometry (ground and space). Living Reviews in Relativity, 3:3, 2000.

[63] B. S. Sathyaprakash and B. F. Schultz. Physics, astrophysics and cosmology with gravitational waves. Living Reviews in Relativity, 2009.

[64] E.´ E.´ Flanagan and S. A. Hughes. Measuring gravitational waves from binary black hole coalescences. i. signal to noise for inspiral, merger, and ringdown. Physical Review D, 57(8):4535–4565, 1998.

[65] S. W. Bruenn, K. R. De Nisco, and A. Mezzacappa. General relativistic eﬀects in the core collapse supernova mechanism. Astrophys.J., 560:326, 2001.

[66] K. Kotake, W. Iwakami, N. Ohnishi, and S. Yamada. Stochastic nature of gravitational waves from supernova explosions with standing accretion shock instability. Astrophysical Journal, 697:133–136, 2009.

[67] J. W. Murphy and A. Burrows. Criteria for core-collapse supernova explosions by the neutrino mechanism. Astrophysical Journal, 688:1159–1175, 2008.

111 [68] T. Takiwaki, K. Kotake, S. Nagataki, and K. Sato. Magneto-driven shock waves in core-collapse supernova. Astrophys.J., 616:1086–1094, 2004.

[69] Y. Suwa, K. Kotake, T. Takiwaki, S. C. Whitehouse, M. Liebendoerfer, and K. Sato. Explosion geometry of a rotating 13 m star driven by the sasi-aided neutrino-heating supernova mechanism. Astron. Soc. Jap., 62:49–53, 2010.

[70] A. Burrows, E. Livne, L. Dessart, C. D. Ott, and J. Murphy. A new mechanism for core-collapse supernova explosions. Astrophys.J., 640:878–890, 2006.

[71] S. W. Bruenn, A. Mezzacappa, W. R. Hix, J. M. Blondin, P. Marronetti, O. E. B. Messer, C. J. Dirk, and S. Yoshida. 2d and 3d core-collapse supernovae simulation results obtained with the chimera code. Journal of Physics: Conference Series, 180:012018, 2009.

[72] S. W. Bruenn, C. J. Dirk, A. Mezzacappa, J. C. Hayes, J. M. Blondin, W. R. Hix, and O. E. B. Messer. Modeling core collapse supernovae in 2 and 3 dimensions with spectral neutrino transport. J.Phys.Conf.Ser., 46:393–402, 2006.

[73] http://wonka.physics.ncsu.edu/pub/vh-1/index.php.

[74] P. Colella and Paul R. Woodward. The piecewise parabolic method (ppm) for gas-dynamical simulations. Journal of Computational Physics, 54:174–201, 1984.

[75] Abu Salah M. Ahsan. Subjecting the chimera supernova code to two hydrodynamic test problems, (i) riemann problem and (ii) point blast explosion. Master’s thesis, Florida Atlantic University, 2008.

[76] R. S. Sutherland, D. K. Bisset, and G. V. Bicknell. The numerical simulation of radiative shocks. i. the elimination of numerical shock instabilities using a local oscillation ﬁlter. Astrophysical Journal, 147:187, 2003.

[77] J. M. Blondin and Peter Lundqvist. Formation of the circumstellar shell around sn 1987a. Astrophysical Journal, 405:337–352, 1993.

[78] A. Marek, H. Dimmelmeier, H.-Th. Janka, E. Mueller, and R. Buras. Exploring the relativistic regime with newtonian hydrodynamics: An improved eﬀective gravitational potential for supernova simulations. Astron.Astrophys., 445:273, 2006.

[79] Ewald M¨ullerand Matthias Steinmetz. Simulating self-gravitating hydrodynamic ﬂows. Computer Physics Communications, 89(1-3):45–58, 1995.

[80] J. R. Oppenheimer and G. M. Volkoﬀ. On massive neutron cores. Physical Review, 55:374–381, 1939.

112 [81] R. C. Tolman. Static solutions of einstein’s ﬁeld equations for spheres of ﬂuid. Physical Review, 55:364–373, 1939.

[82] M. Liebendoerfer, A. Mezzacappa, and F.-K. Thielemann. Conservative general relativistic radiation hydrodynamics in spherical symmetry and comoving coordinates. Phys.Rev. D, 63:104003, 2001.

[83] A. Mezzacappa and S. W. Bruenn. A numerical method for solving the neutrino boltzmann equation coupled to spherically symmetric stellar core collapse. Astrophysical Journal, 405:669–684, 1993.

[84] M. Liebendoerfer, O. E. B. Messer, A. Mezzacappa, S. W. Bruenn, C. Y. Cardall, and F.-K. Thielemann. A ﬁnite diﬀerence representation of neutrino radiation hydrodynamics in spherically symmetric general relativistic space-time. Astrophys.J.Suppl., 150:263–316, 2004.

[85] R. Buras, M. Rampp, H.-Th. Janka, and K. Kifonidis. Two-dimensional hydrodynamic core-collapse supernova simulations with spectral neutrino transport. i. numerical method and results for a 15 solar masses star. Astron.Astrophys., 447:1049–1092, 2006.

[86] S. Reddy, M. Prakash, and J. M. Lattimer. Neutrino interactions in hot and dense matter. Physical Review D, 58(1):013009, 1998.

[87] S. Hannestad and G. G. Raﬀelt. Supernova neutrino opacity from nucleon-nucleon bremsstrahlung and related processes. Astrophysical Journal, 507(1):339–352, 1998.

[88] C. Hanhart, J. A. Pons, and Sanjay Reddy. The likelihood of gods’ existence: improving the sn 1987a constraint on the size of large compact dimensions. Physics Letters B, 509:1–9, 2001.

[89] W. R. Hix and F.-K. Thielemann. Silicon burning. ii. quasi-equilibrium and explosive burning. Astrophysical Journal, 511:862–875, 1999.

[90] W. R. Hix and B. S. Meyer. Thermonuclear kinetics in astrophysics. Nuclear Physics A, 777:188–207, 2006.

[91] W. D. Arnett and J. W. Truran. Carbon-burning nucleosynthesis at constant temperature. Astrophysical Journal, 157:339, 1969.

[92] T. Rauscher and F.-K. Thielemann. Astrophysical reaction rates from statistical model calculations. Atomic Data and Nuclear Data Tables, 75:1–351, 2001.

[93] J. M. Lattimer and D. F. Swesty. A generalized equation of state for hot, dense matter. Nuclear Physics A, 535(2):331, 1991.

113 [94] J. Cooperstein. The equation of state in supernovae. Nuclear Physics A, 438(3):722, 1985. [95] F.-K. Thielemann, K. Nomoto, and K. Yokoi. Explosive nucleosynthesis in carbon deflagration models of type i supernovae. Astron. Astrophys., 158:17– 33, 1986. [96] S. Nagataki, M. Hashimoto, K. Sato, and S. Yamada. Explosive nucleosynthesis in axisymmetrically deformed type ii supernovae. Astrophysical Journal, 486:1026, 1997. [97] Shin’ichirou Yoshida. Personal communication, 2008. [98] R. Epstein. Post-newtonian generation of gravitational waves. Astrophysical Journal, 197:717, 1975. [99] T. Nakamura and K. Oohara. Frontiers in Numerical Relativity. Cambridge University Press, Cambridge, 1989. [100] L. Blanchet, T. Damour, and G. Schaefer. Post-newtonian hydrodynamics and post-newtonian gravitational wave generation for numerical relativity. Royal Astronomical Society, Monthly Notices, 242:289–305, 1990. [101] L. S. Finn and C. R. Evans. Determining gravitational radiation from newtonian self-gravitating systems. Astrophysical Journal, 351:588, 1990. [102] R. Epstein. The generation of gravitational radiation by escaping supernova neutrinos. Astrophysical Journal, 223:1037, 1978. [103] E. Mueller and H.-Th. Janka. Gravitational radiation from convective instabilities in type ii supernova explosions. Astronomy and Astrophysics, 317:140–163, 1997. [104] S. E. Woosley and A. Heger. Nucleosynthesis and remnants in massive stars of solar metallicity. Physics Reports, 442:269–283, 2007. [105] S. E. Woosley and T. Weaver. The evolution and explosion of massive stars. ii. explosive hydrodynamics and nucleosynthesis. Astrophysical Journal, 101:181, 1995. [106] W. Hillebrandt, K. Nomoto, and R. G. Wolff. Supernova explosions of massive stars - the mass range 8 to 10 solar masses. Astron. Astrophys., 133:175, 1984. [107] J. C. Hayes, M. Norman, R. Fieldler, J. Bordner, P. S. Li, S. Clark, A. ud- Doula ud-Doula ud-Doula ud-Doula ud-Doula ud-Doula ud-Doula ud-Doula ud-Doula ud Doula, and M.-M. MacLow. Simulating radiating and magnetized flows in multiple dimensions with zeus-mp. Astrophysical Journal Supplement Series, 165:188, 2006.

114 [108] S. Ando, J. F. Beacom, and H. Yuksel. Detection of neutrinos from supernovae in nearby galaxies. Phys.Rev.Lett., 95:171101, 2005.

[109] H.-Th. Janka, A. Marek, B. Mueller, and L. Scheck. Supernova explosions and the birth of neutron stars. AIP Conf.Proc., 983:369–378, 2008.

[110] E. Mueller, H.-Th. Janka, and A. Wongwathanarat. Parametrized 3d models of neutrino-driven supernova explosions, July 2011.

[111] Anthony Mezzacappa. The core collapse supernova mechanism: Current models, gaps, and the road ahead, October 2004.

[112] T. Nakamura, K. Oohara, and Y. Kojima. General relativistic collapse to black holes and gravitational waves from black holes. Prog. Theor. Phys. Suppl., 90:1–218, 1987.

[113] M. Shibata and T. Nakamura. Evolution of three-dimensional gravitational waves: Harmonic slicing case. Phys. Rev. D, 52:5428, 1995.

[114] T. W. Baumgarte and S. L. Shapiro. Numerical integration of einstein’s ﬁeld equations. Phys. Rev. D, 59:024007, 1999.

[115] T. Zwerger and E. Mueller. Dynamics and gravitational wave signature of axisymmetric rotational core collapse. Astron. Astrophys., 320:209, 1997.

[116] K. Kotake, S. Yamada, and K. Sato. Gravitational radiation from axisymmetric rotational core collapse. Phys.Rev.D, 68:044023, 2003.

[117] M. Shibata and Y.-I. Sekiguchi. Gravitational waves from axisymmetric rotating stellar core collapse to a neutron star in full general relativity. Phys. Rev. D, 69:084024, 2004.

[118] C. D. Ott, A. Burrows, E. Livne, and R. Walder. Gravitational waves from axisymmetric, rotational stellar core collapse. Astrophys.J., 600:834–864, 2004.

[119] H. Dimmelmeier, C. D. Ott, H.-Th. Janka, A. Marek, and E. Mueller. Generic gravitational wave signals from the collapse of rotating stellar cores. Phys.Rev.Lett., 98:251101, 2007.

[120] S. Scheidegger, T. Fischer, S.C. Whitehouse, and M. Liebendoerfer. Grav- itational waves from 3d mhd core collapse simulations. Astron. Astrophys., 490:231, 2008.

[121] A. Heger, S. E. Woosley, and H. C. Spruit. Presupernova evolution of diﬀeren- tially rotating massive stars including magnetic ﬁelds. Astrophysical Journal, 626:350, 2005.

115 [122] C. D. Ott, A. Burrows, T. A. Thompson, E. Livne, and R. Walder. The spin periods and rotational proﬁles of neutron stars at birth. Astrophys.J.Suppl., 164:130–155, 2006.

[123] S. D. Mohanty, Sz. Marka, R. Rahkola, et al. Gamma ray bursts and gravitational waves: triggered search strategy in the ligo science run. Class. Quantum Grav., 21:765–774, 2004.

[124] S. A. Balbus and J. F. Hawley. Instability, turbulence, and enhanced transport in accretion disks. Reviews of Modern Physics, 70:1–53, 1998.

[125] S. E. Woosley and J. S. Bloom. The supernova gamma-ray burst connection. Astronomy, 44:507–556, 2006.

116