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Pos(HEASA2017)001 ∗ N.Bishop@Ru.Ac.Za Speaker Current status of gravitational wave observations PoS(HEASA2017)001 Nigel T. Bishop∗ Rhodes University E-mail: [email protected] We review LIGO and VIRGO detection events reported so far, as well as future prospects. We also review the prospects for gravitational wave measurements, but in lower frequency bands, from pulsar timing arrays, and from the space system LISA. 5th Annual Conference on High Energy Astrophysics in Southern Africa 4-6 October, 2017 University of the Witwatersrand (Wits), South Africa ∗Speaker. c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ Gravitational wave observations Nigel T. Bishop 1. Introduction Advanced LIGO has completed its first observation runs (O1: 2015.09.29 to 2016.01.12, O2: 2016.11.30 to 2017.08.25); Virgo has also come on line and completed its first observation run 2017.07.28 to 2017.08.25. To date, there have been reported 5 definite and 1 probable black hole mergers, GW150914, GW151226, GW170104, GW170608, GW170814 and LVT151012; and one neutron star merger event, with an electromagnetic counterpart, GW170817. Other sig- nals that were possibly expected, have not been observed, including burst events such as Galactic supernovas and cosmic string interactions, and continuous sources caused by the rotation of a non- PoS(HEASA2017)001 axisymmetric body examples being the Crab, Vela and other pulsars. Even such non-detections provide constraints on astrophysical models. The limited results obtained so far have already had an impact on theoretical astrophysics, since an expalanation for the origin of black holes with mass O(30M ) is required. As will be discussed later, an enormous increase in detections is expected over the next few years. Thus gravitational wave (GW) observations will provide strong constraints on the modeling of various astrophysical and physical processes. This review starts with an introduction to the basic theory of GW generation and propagation; then summarizes the functioning of a LIGO machine; the events that have been detected to date are described, as well as the implications for physics and astrophysics. We then look at future prospects, in terms of the current LIGO facilities as well as planned ones; and other plans for GW detection, i.e. the satellite system LISA and pulsar timing arrays. 2. GW basics The simplest approach to GWs is to linearize Einstein’s equations, usually about a flat Minkowski background, and it is found that, to leading order, the GW field takes the form 2 2 2 2 2 ds = −dt + dx + (1 + h+(t;x))dy + (1 − h+(t;x))dz + 2h×(t;x)dydz; (2.1) using Cartesian coordinates (t;x;y;z) for a plane GW propagating in the x-direction. The functions 2 2 h+(t;x);h×(t;x) are independent and each satisfies the wave equation (¶t − ¶x = 0). They con- stitute the two polarization modes of the GW; if the (x;y) axes are rotated through p=4 about the z-axis, then the two modes are interchanged. A GW detector comprises two perpendicular arms each of length L, and laser light travels along each arm and back again. The difference in travel time between the two arms is measured as an interference fringe; see Fig. 1. For a more detailed discussion of the operation of a LIGO detector, see for example [1]. If, for a given wave, the arms lie along the y and z axes, then 2Lh+ is measured; whereas if the arms are in the y;z plane but at p=4 to the axes, then 2Lh× is meaured. If one of the arms is in the direction of propagation of the wave, then its length will not change; but the other arm will be transverse to the wave, and the relative change of length will be half that of the case where both arms are transverse to the wave. The one case in which no effect will be recorded is if the wave is linearly polarized, which means that either h+ or h× is zero, and the arms are oriented such that the zero component is measured. The current LIGO detectors have arms of length 4km, and for VIRGO the length is 3km. This 1 Gravitational wave observations Nigel T. Bishop PoS(HEASA2017)001 Figure 1: Illustration of the schematics of the LIGO facilities, adapted from [2] leads to the detectors being sensitive only to signals in the frequency range approximately 20Hz to 2000Hz. The generation of GWs by a source is often described by Einstein’s quadrupole formula [3, 4], which is valid for systems where the relative velocities are small. For a binary (see Fig. 2) comprising two masses, M1 and M2 in circular orbit of diameter r0 and with angular velocity w, the formula gives [5] 2 2 8M1M2r0w 2 (h+;h×) = − × (1 + cos q)cos(2wt − 2f); 2cosq sin(2wt − 2f) ; (2.2) (M1 + M2)r indicating that the wave frequency is twice that of the orbital frequency. The wave magnitude falls off as 1=r. Thus, if detection sensitivity increases by a factor 2, a source could be 2 times further, so the volume searched and consequently the expected event detection rate would be 23 = 8 times larger. The energy carried away by GWs is I dE 1 2 2 = − (¶t h+) + (¶t h×) ; (2.3) dt 4p S 2 Gravitational wave observations Nigel T. Bishop PoS(HEASA2017)001 Figure 2: Two masses, M1 and M2 in circular orbit diameter 2r0 rotating at angular velocity w. where S is a spherical shell at large, constant r. (Actually, the energy emitted has historically been a controversial issue, which was resolved only in 1962 by Bondi et al. [6].) The rate of energy emission of GWs is equated to the rate of loss of Newtonian orbital energy, yielding 2 2 dE M1 M2 (M1 + M2) d M1M2 M1M2 = − 5 = − = ¶t r0 2 : (2.4) dt 5r0 dt 4r0 4r0 Applying the relation between w and r0 using Newtonian orbit theory (Kepler’s law), we find s M1 + M2 ¶t w 3 ¶t r0 w = 3 ; = − ; (2.5) 8r0 w 2 r0 −2=3 so that r0 ∝ w . We can now use Eq. (2.5) in Eq. (2.4) to replace ¶t r0 and r0 by ¶t w and w to obtain the chirp mass formula (M M )3=5 5 3=5 ≡ 1 2 = −8=3 f −11=3( f ) ; (2.6) M 1=5 p ¶t (M1 + M2) 96 where f is the wave frequency, related to the orbital angular velocity by w = p f . The key point is that both f and ¶t f are determinable entirely from LIGO data, and thus so is the chirp mass. 3 Gravitational wave observations Nigel T. Bishop Observed changes to the orbital frequency of the Hulse-Taylor binary pulsar, and other systems, have provided confirmation of this formula since the 1970s. We use Kepler’s law Eq. (2.5) to replace r0 by f in Eq. (2.2), and find the wave amplitude 2M 5=3(p f )2=3 (A ; A ) = (1 + cos2 q); 2cosq ; (2.7) + × r where (h+;h×) = (A+ cos(2wt − 2f);A× sin(2wt − 2f)). Thus, once the chirp mass is determined and if the polarization modes can be separated so that the inclination q of the source as seen by the detector is constrained, then the above result may be used to constrain the distance to the source. PoS(HEASA2017)001 The redshift z of the source causes frequencies in the source frame to be higher than in the detector frame by a factor (1 + z), and this gives a correction that source frame masses are smaller than detector frame masses by a factor 1=(1+z). In the absence of an optical counterpart and direct measurement, the redshift is estimated from the distance using Hubble’s law. Qualitatively, the system evolves with the orbital diameter slowly decreasing, and the wave magnitude, wave frequency and orbital velocity slowly increasing. Once the orbital velocity (i.e. r0w) is not negligible compared to the velocity of light, then alternative methods need to be used for accurate waveform estimation; this is essential, since the wave amplitude is largest, and therefore more detectable, in this regime. These methods may be analytic (post-Newtonian, post- Minkowskian and self-force), or they may require numerical evolution of the Einstein equations; but they are not discussed further in this review. However, the quadrupole formula gives results that are accurate as order of magnitude estimates even outside its domain of applicability. Except, possibly, in the case of two neutron stars, the end product will be a black hole, and black hole perturbation theory [7] can be used to estimate the post-merger waveform. This gives t wt M M (h+;h×) ∼ exp(− )cos( ); where t = 0:056 ms; w = 12000 Hz; (2.8) t 2p M M and where M is the total mass of the system. The amplitude and frequency increase until they reach maxima at merger. The maximum frequency is of order M fmax ≈ 5000Hz; (2.9) m where m = M1M2=(M1 + M2) is the reduced mass of the system Thus LIGO would not see events with M1;M2 ' 500M since throughout the inspiral and merger the wave frequency would be too low. Let fin be the frequency when the signal is first “in band”; normally, fin ≈ 20Hz, but if the signal is weak it could be larger. Integrating the chirp formula Eq. (2.6), we find that the signal is viewed in the detector for a time 5 t = ; (2.10) 8=3 5=3 8=3 256p M fin where the integration is taken until f ! ¥ representing merger.
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