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Einstein Polarizations of Continuous Gravitational Waves Isi, M., Weinstein, A. J., Mead, C., and Pitkin, M. (2015) Detecting beyond- Einstein polarizations of continuous gravitational waves. Physical Review D, 91, 082002. Copyright © 2015 American Physical Society. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge The content must not be changed in any way or reproduced in any format or medium without the formal permission of the copyright holder(s) When referring to this work, full bibliographic details must be given http://eprints.gla.ac.uk/104666/ Deposited on: 22 April 2015. Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk LIGO-P1400169 Detecting Beyond-Einstein Polarizations of Continuous Gravitational Waves Maximiliano Isi,1, ∗ Alan J. Weinstein,1, y Carver Mead,1, z and Matthew Pitkin2, x 1California Institute of Technology 2University of Glasgow (Dated: March 29, 2015) The direct detection of gravitational waves with the next generation detectors, like Advanced LIGO, provides the opportunity to measure deviations from the predictions of General Relativity. One such departure would be the existence of alternative polarizations. To measure these, we study a single detector measurement of a continuous gravitational wave from a triaxial pulsar source. We develop methods to detect signals of any polarization content and distinguish between them in a model independent way. We present LIGO S5 sensitivity estimates for 115 pulsars. I. INTRODUCTION which make use of GW burst search methods [12]. In this paper, we present methods to search LIGO{like Since its introduction in 1915, Einstein's theory of detector data for continuous GW signals of any polariza- General Relativity (GR) has been confirmed by exper- tion mode, not just those allowed by GR. We also com- iment in every occasion [1]. However, GR has not yet pare the relative sensitivity of different model{dependent been tested with great precision on scales larger than the and independent templates to certain kinds of signals. solar system or for highly dynamical and strong gravita- Furthermore, we provide expected sensitivity curves for tional fields [2]. Those kinds of rapidly changing fields GR and non{GR signals, obtained by means of blind give rise to gravitational waves (GWs)|self propagat- searches over LIGO noise (not actual upper limits). ing stretching and squeezing of spacetime originating in Section II provides the background behind GW polar- the acceleration of massive objects, like spinning neu- izations and continuous waves, while sections III and V tron stars with an asymmetry in their moment of inertia present search methods and the data analysis procedures (e.g., see [3, 4]). used to evaluate sensitivity for detection. Results and fi- Although GWs are yet to be directly observed, detec- nal remarks are provided in sections V & VI respectively. tors such as the Laser Interferometer Gravitational Wave Observatory (LIGO) expect to do so in the coming years, giving us a chance to probe GR on new grounds [5, 6]. II. BACKGROUND Because GR does not present any adjustable parameters, these tests have the potential to uncover new physics [1]. A. Polarizations By the same token, LIGO data could also be used to test alternative theories of gravity that disagree with GR on Just like electromagnetic waves, GWs can present dif- the properties of GWs. ferent kinds of polarizations. Most generally, metric the- Furthermore, when looking for a weak signal in noisy ories of gravity could allow six possible modes: plus (+), LIGO data, certain physical models are used to target cross ( ), vector x (x), vector y (y), breathing (b) and the search and are necessary to make any detection pos- longitudinal× (l). Their effects on a free{falling ring of sible [2]. Because these are usually based on predictions particles are illustrated in fig. 1. Transverse GWs (+, from GR, assuming an incorrect model could yield a weak and b) change the distance between particles separated in× detection or no detection at all. Similarly, if GR is not the plane perpendicular to the direction of propagation a correct description for highly dynamical gravity, check- (taken to be the z-axis). Vector GWs are also transverse; ing for patterns given by alternative models could result but, because all particles in a plane perpendicular to the in detection where no signal had been seen before. direction of propagation are equally accelerated, their There exist efforts to test GR by looking at the de- relative separation is not changed. Nonetheless, parti- viations of the parametrized post{Newtonian coefficients cles farther from the source move at later times, hence extracted from the inspiral phase of compact binary co- varying their position relative to points with both dif- alescence events [7{9]. Besides this, deviations from GR ferent x{y coordinates and different z distance. Finally, could be observed in generic GW properties such as po- longitudinal GWs change the distance between particles larization, wave propagation speed or parity violation separated along the direction of propagation. [1, 10, 11]. Tests of these properties have been proposed Note that, because of their symmetries, the breathing and longitudinal modes are degenerate for LIGO-like in- terferometric detectors, so it is enough to just consider ∗ [email protected] one of them in the analysis. Also, this study assumes y [email protected] wave frequency and speed remain constant across modes, z [email protected] which restricts the detectable differences between polar- x [email protected] izations to amplitude modulations. 2 y x y odic motion, such as binary systems or spinning neutron stars [14]. Throughout this paper, we target known pul- sars (e.g., the Crab pulsar) and assume an asymmetry x z x in their moment of inertia (rather than precession of the spin axis or other possible, but less likely, mechanisms) causes them to emit gravitational radiation. A source of this type can generate GWs only at multiples of its y y y rotational frequency ν. In fact, it is expected that most power be radiated at twice this value [15]. For that rea- son, we take the GW frequency, νgw, to be 2ν. Moreover, x z z the frequency evolution of these pulsars is well{known thanks to electromagnetic observations, mostly at radio wavelengths but also in gamma-rays. Simulation of a CGW from a triaxial neutron star is FIG. 1: Illustration of the effect of different GW straightforward. The general form of a such signal is: polarizations on a ring of test particles. plus (+) and cross ( ) tensor modes (green); vector{x (x) and vector{y× (y) modes (red); breathing (b) and longitudinal (l) scalar modes (black). In all of these X h(t) = A (t; α; δ; λ, φ, γ; ξ) h (t; ι, h ; φ ; ν; ν;_ ν¨); diagrams the wave propagates in the z{direction [1]. p p 0 0 p j (1) where, for each polarization p, Ap is the detector response In reality, however, GWs might only possess some of (antenna pattern) and hp a sinusoidal waveform of fre- those six components: different theories of gravity pre- quency νgw = 2ν. The detector parameters are: λ, lon- dict the existence of different polarizations. In fact, due gitude; φ, latitude; γ, angle of the detector x{arm mea- to their symmetries, + and are associated with tensor sured from East; and ξ, the angle between arms. Values × theories, x and y with vector theories, and b and l with for the LIGO Hanford Observatory (LHO), LIGO Liv- scalar theories. In terms of particle physics, this differ- ingston Observatory (LLO) and Virgo (VIR) detectors entiation is also linked to the predicted helicity of the are presented in table I. The source parameters are: , graviton: 2, 1 or 0, respectively. Consequently, GR the signal polarization angle; ι, the inclination of the pul- ± ± only allows + and , while scalar{tensor theories also sar spin axis relative to the observer's line-of-sight; h , × 0 predict the presence of some extra b component whose an overall amplitude factor; φ0, a phase offset; and ν, the strength depends on the source [1]. Bolder theories might rotational frequency, withν _,ν ¨ its first and second deriva- predict the existence of vector or scalar modes only, while tives. Also, α is the right ascension and δ the declination still being in agreement with all other non{GW tests. of the pulsar in celestial coordinates. Four-Vector Gravity (G4v) is one such extreme exam- ple [13]. This vector{based framework claims to repro- Note that the inclination angle ι is defined as is stan- duce all the predictions of GR, including weak–field tests dard in astronomy, with ι = 0 and ι = π respectively and total radiated power of GWs. However, this theory meaning that the angular momentum vector of the source differs widely from GR when it comes to gravitational points towards and opposite to the observer. The signal wave polarizations. Thus, one of the only ways to test polarization angle is related to the position angle of G4v would be to detect a GW signal composed of x and the source, which is in turn defined to be the East angle y modes instead of + and . × of the projection of the source's spin axis onto the plane of the sky. B. Signal Although there are hundreds of pulsars in the LIGO band, in the majority of cases we lack accurate mea- Because of their persistence, continuous gravitational surements of their inclination and polarization angles. waves (CGWs) provide the means to study GW polar- The few exceptions, presented in table II, were obtained izations without the need for multiple detectors. For the through the study of the pulsar spin nebula [16].
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