Springer Theses Recognizing Outstanding Ph.D. Research Pantelis Pnigouras Saturation of the f -mode Instability in Neutron Stars Springer Theses Recognizing Outstanding Ph.D. Research Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected foritsscientificexcellenceandthehighimpactofitscontentsforthepertinentfield of research. For greater accessibility to non-specialists, the published versions includeanextendedintroduction,aswellasaforewordbythestudent’ssupervisor explainingthespecialrelevanceoftheworkforthefield.Asawhole,theserieswill provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. 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More information about this series at http://www.springer.com/series/8790 Pantelis Pnigouras Saturation of the f-mode Instability in Neutron Stars Doctoral Thesis accepted by ü ü the Eberhard-Karls University of T bingen, T bingen, Germany 123 Author Supervisor Dr. Pantelis Pnigouras Prof. KostasD.Kokkotas MathematicalSciences andSTAG Research Eberhard-Karls University of Tübingen Centre Tübingen,Germany University of Southampton Southampton, UK ISSN 2190-5053 ISSN 2190-5061 (electronic) SpringerTheses ISBN978-3-319-98257-1 ISBN978-3-319-98258-8 (eBook) https://doi.org/10.1007/978-3-319-98258-8 LibraryofCongressControlNumber:2018950792 ©SpringerNatureSwitzerlandAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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To my mother, for relishing the voyage. ’ Supervisor s Foreword The detection of gravitational waves in 2015 signalled a new era for theoretical physics and astrophysics. Thanks to the advanced technology developed, the dark sideoftheUniverseisbecomingmoretransparenttohumans,withblackholesand neutron stars revealing their violent faces in both the gravitational and the elec- tromagneticspectrum.ItwillalsonotbelonguntilthemysteryoftheBigBangwill be unveiled via gravitational waves. Last year, for the first time in the history of astronomy,aviolentevent—themergingoftwoneutronstars—wasobservedinall bands of the electromagnetic spectrum, following its discovery via gravitational- wave emission. There are many fundamental questions which can be tackled with neutron star observations.Twoofthemostcompellingonesrelatefirstofalltothedynamicsof neutron stars and second to their efficiency as gravitational-wave sources. By answeringthesetwoquestions,weexpecttorevealtheequationofstateofmatterat supranuclear densities for the first time. Typical, fast-rotating neutron stars, with masses 1–2 times larger than the solar mass,areexpectedtobetheoutcomeofthegravitationalcollapseofamassivestar. Furthermore, themerging of twosuchneutronstarswill genericallyproduce, most of the times, a supermassive neutron star. The newly formed object (born from either of the above two formation channels) will be hot and will rotate fast until, possibly,themagneticfieldbrakesitsrotationdowntothespinsthatweobservein the known, old neutron stars. During this early stage, rotational instabilities may take place, deforming the nascent object and emitting copious amounts of gravi- tational waves at specific frequencies, which will carry information about the physical parameters that describe neutron stars. One of the oscillation modes that may become unstable due to rotation is the so-called f-mode, which is associated with the fundamental oscillation frequency. Theunstablepatternsofpulsationofthef-modegrowintimeand,whenaspecific amplitudeisreached,theysaturatebytransferringoscillationenergytoothermodes via nonlinear mode coupling. The saturation amplitude of the f-mode is directly related to the efficiency with which gravitational waves are emitted. Since 2002 there have been systematic studies about the saturation of another potentially ix x Supervisor’sForeword unstablemode,theso-calledr-mode.But,untilnow,therehadbeennosuchstudy for the saturation of the f-mode instability. A plausible reason could be the com- plicated nature of the problem, compared to similar studies for the r-mode. This thesis is the first systematic and successful attempt in addressing this question. It includes the basic theory of nonlinear mode coupling and develops the method- ology for the f-mode. Finally, it arrives at concrete estimations by applying the above to the two promising astrophysical sources of gravitational waves, namely typical and supramassive neutron stars, and examines the possibility of observing the cosmological stochastic background of gravitational waves due to f-mode instabilities throughout the Universe. The most important application of this work, which is extremely well-timed, is relatedtothepost-mergersupramassiveneutronstarbornfromthecollision,which, apart from its large mass, acquires extremely high spin. This combination favours the onset of rotational instabilities and the neutron star can become unstable for a quite wide range of temperatures and spins. In addition, the instability grows on very short timescales, of the order of a few seconds, during which even the strongest magnetic field will not be able to drain significant amounts of angular momentum. This scenario was studied in detail in Doneva et al. (2015), and these initial results were very promising. The existence of this specific “gravitational-wave afterglow” can be correlated with the observed light curves of short (cid:1)-ray bursts, which, in many cases, acquire a plateau lasting hundreds to thousands of seconds and suggesting the survival of the post-merger neutron star remnant for minutes to hours before collapsing to a black hole. During this phase, thestarisunstableandtheemittedgravitationalradiationshouldbedetectableupto a few tens of Mpc with current gravitational-wave detectors, Advanced LIGO and Virgo,anduptoafewhundredsofMpcwiththeplannednext-generationdetectors, the Einstein Telescope and the Cosmic Explorer. I believe that the thesis includes an excellent review on oscillations and insta- bilities of neutron stars that can be pleasantly read by anyone, while in the appendices one will find the detailed analytic calculations and the extensive for- mulae derived for the problem. Concluding, I consider Dr. Pnigouras’s thesis an excellent piece of scientific work,writteninanelegant,inspiring,andeasy-to-readway.Theresultsaresound, timely, and came out of a combination of analytical and computational work. Tübingen, Germany Prof. Kostas D. Kokkotas June 2018 Reference Doneva,D.D.,Kokkotas,K.D.&Pnigouras,P.(2015).Gravitationalwaveafterglowinbinary neutron star mergers. Physical Review D, 92, 104040. https://doi.org/10.1103/PhysRevD.92. 124004,arXiv:1510.00673. Preface Sincetheirtheoreticalpredictionin1934andtheserendipitousdiscoveryofthefirst pulsar in 1967, neutron stars remain among the most challenging objects in the Universe. Thanks to the advancement of theory, experiments, and observations, many aspects of their nature have been deciphered, yet their inner structure is still unknown. Gravitational waves emitted by neutron star oscillations can be used to obtain information about their equation of state, that is, the equation of state of dense nuclearmatter. As discovered in the1970s, certain oscillation modes can be secularly unstable to the emission of gravitational radiation, via the so-called Chandrasekhar-Friedman-Schutz (CFS) mechanism, thus rendering gravitational- wave asteroseismology a promising probe of the neutron star interior, especially after the recent birth of gravitational-wave astronomy. After its initial growth phase, the instability is expected to saturate, due to nonlinear effects. The saturation amplitude of the unstable mode determines the detectability of the generated gravitational-wave signal, but also affects the evo- lution of the neutron star through the instability window, namely the region where the instability is active. In this work, we study the saturation of CFS-unstable f- modes(fundamentalmodes),duetolow-ordernonlinearmodecoupling.Usingthe quadratic-perturbation approximation, we show that the unstable (parent) mode resonantly couples to pairs of stable (daughter) modes, which drain the parent’s energy and make it saturate, via a mechanism called parametric resonance insta- bility. The saturation amplitude of the most unstable f-mode multipoles is calcu- lated throughout their instability windows, for typical and supramassive newborn neutron stars, simply modelled as polytropes in a Newtonian context. Contrary to previous studies, where the saturation amplitude is treated as a constant, we find that it changes significantly throughout the instability window and,hence,duringtheneutronstarevolution.Usingthehighestvaluesobtainedfor thesaturationamplitude,asignalfroman unstable f-modemay evenlie abovethe sensitivity of current, second-generation, gravitational-wave detectors. InChap.1,wepresentabriefhistoryofthefieldandthereasonswhichmotivate suchanenterprise,startingfromtheconceptofasteroseismologyandhowitcanbe applied in neutron stars, so that the equation of state of dense nuclear matter is xi