Terrestrial Reference System and Elastic, Non-rotating Earth ModeL

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Detectors and Frequency Ranges

Detection of the GW has many diﬀerent approaches for diﬀerent part of the GW spec-trum. Each frequency band is characterized by diﬀerent operating detectors and diﬀerent gravitational sources. There are four frequency bands of interest:
• extremely low frequency (ELF) band from ∼ 10−15 to ∼ 10−18 Hz.
• very low frequency (VLF) band from ∼ 10−7 to ∼ 10−9 Hz.
• low frequency (LF) band from ∼ 10−4 to ∼ 0.1 Hz.
• high frequency (HF) band from ∼ 10 to ∼ 103 Hz.
In the ELF band the GWs are sought via their imprint on the polarization of the cosmic microwave background (CMB) radiation, a relic of the early Universe. The expected sources are the quantum fluctuations in the gravitational field (space-time curvature) that emerged from the Big Bang’s event (Thorne and Blandford, 2017).
In the VLF band radio astronomers search for small irregularities in the arrival times of the pulsar signals caused by the GWs. Pulsars are spinning neutron stars that emit strong radio waves due to their rotational energy. Their very high rotational energy generates electric field from the movement of the very strong magnetic field, which result in the accel-eration of protons and electrons on the star surface and thus creation of the electromagnetic beam emanating from the poles of the magnetic field. Every time a magnetic pole points toward the Earth, the beamed emission is observed as a ’pulse’ of radio waves. Therefore, each pulsar act as a regular clock. The measured pulse arrival time can be compared against a prediction, where residual times include the eﬀects of the passing GW through the radio array (Schutz, 2009). Current pulsar timing arrays (PTAs) in operation are the European Pulsar Timing Array (EPTA1; Kramer and Champion (2013); the detector sensitivities in Babak et al. (2016)), the Parkes Pulsar Timing Array (PPTA2; Hobbs (2013)) in Australia and the North American Nanohertz Observatory for Gravitational Waves (NANOGrav3, McLaughlin (2013)). The International Puslar Timing Array (IPTA4; Manchester (2013)) is a consortium of consortia, comprised of EPTA, PPTA, NANOGrav, and its principal goal is to detect GWs using one single array of approximately 30 pulsars. The next de-tector prototype in the radio astronomy is the Square Kilometre Array (SKA5; Dewdney et al. (2009)), the world’s largest radio telescope. It will be made up of arrays of antennas – SKA-mid observing mid to high frequencies and SKA-low observing low frequencies – spread approximately over one square kilometre. Having the receiving stations extending over the vast area it would provide the highest resolution images in all astronomy. In the LF band a common method to detect GWs is interplanetary spacecraft tracking. This technique works on the principle of a general beam detector (Thorne and Blandford, 2017). Spacecraft carries transponders, radio receivers, that amplify and return the signal from the ground tracking stations. A measurement of the return time defines the position of the spacecraft and if this measurement is accurate enough, small changes in the return time of the radio signals might indicate existence of the GWs (Armstrong, 2006). Since it turned out that the sensitivity of these searches is not high, this technique will be supplanted by Laser Interferometer Space Antenna (LISA6, Bender et al. (1995)) and eLISA7 (Amaro-Seoane et al., 2012), rescoped version of the classic LISA mission. LISA is a space-based detector that consists of three satellites flying in a triangular constellation with arms of length 5 × 109 m in a 1 AU orbit around the Sun, trailing the Earth by 20◦. Each satellite contains of two telescopes, two lasers and two free falling test masses arranged so that each satellite point at the other two. This forms Michelson-like interferometers, each centered on one of the satellites, with the test masses defining the ends of the arms. Therefore, LISA-like detectors use laser interferometry, however the laser beams are not contained in cavity and the beam travels only once along each arm, unlike in ground-based detectors. The distances between the satellites are precisely monitored and every distortions will be caused by the passing GWs. Each satellite is a zero-drag satellite, which eﬀectively floats around test masses, maintaining themselves centered around the masses and monitoring their relative position to the spacecraft. Thus, using this principle all non-gravitational forces are eliminated. eLISA (evolved Laser Interferometer Space Antenna) is designed to probe the same frequency range as LISA and the main diﬀerences are the shorter arms (109 m), two laser arms instead of three and diﬀerent orbit. Some of the potential sources for space-based detectors are massive black holes mergers at the centre of galaxies, massive black holes orbited by small compact objects, extreme mass ratio inspirals, binaries of compact stars in our Galaxy. Proposed successor to LISA are the Advanced Laser Interferometer Antenna (ALIA; Crowder and Cornish (2005)), Big Bang Observer (BBO; Crowder and Cornish (2005)) and Deci-hertz Interferometer GW Observatory (DECIGO; Takahashi and Nakamura (2003)). All successors are designed to probe the decihertz region of the GW spectrum. For overview of the GW detection in space the reader is referred to Ni (2016).
In the HF band GWs are detected by the ground-based laser interferometer detectors and the resonant-mass detectors. The laser interferometers utilize the principle of laser interferometry and a common configuration for optical interferometry is the Michelson in-terferometer. It consists minimally of one stable laser (source), a beam splitter (usually a partially reflecting mirror), two reflecting mirrors at the end of two arms and a detector (Fig. 2.1). In Michelson interferometer a single coherent light beam passes through a beam splitter, which sends half the light down one arms and other half down the orthogonal arm. The two beams have correlated phases. In the two arms beams are passing through an optical cavity and are being reflected by the mirrors at the end of the two arms. On their why back they are recombined to an interference pattern measured by detector. Any diﬀerence in the local space time creates a phase diﬀerence between the two beams and this eﬀect is measured by observing the changing interference pattern. Therefore, if two arms have same proper length, beams will return in phase, interfering constructively. Otherwise, beams will return to the detector out of phase and they will interfere destructively. All mirrors and beam splitter are freely floating and suspended in order to filter out the me-chanical vibration noise. The response of the detector to the incident GW depends upon the relative orientations of the detector and the incoming wave. The existed, existing and planned ground-based detector are listed in Tab. (2.1) with relevant references.

Elastic Bodies as Detectors

Around the same time when the first idea about the ground-based detector was established, the idea that GWs could excite the vibrations of elastic bodies, and therefore Earth too, was developed (Pirani, 2009; Weber, 1959). Weber (1959) proposed methods for the interstellar gravitational radiation detection using the fact that the relative motion of mass points are driven by second spatial derivatives of the gravitational fields. He proposed an experiment where the Earth is considered as a block of material representing the GW antenna, a res-onant body, for which the normal modes of the Earth are expected as a response to the excitation. He also discussed generation and detection of GW in the laboratory. Forward et al. (1961) were the first to calculate the upper bound of the GW energy passing through the Earth using the strain data from the seismograph at Isabella, California. They computed the strain magnitude induced by the Riemann tensor in a longitudinally vibrating rod (We-ber, 1959). Next, Weber (1967) provided the first upper limit on the gravitational-radiation flux using a mechanical gravimeter in vicinity of normal mode periods. Tuman (1971, 1973) first claimed a GW detection using the Earth’s normal modes observed in cryogenic gravity meter records. He interpreted a higher energy content in the power spectra of the even harmonic degrees as the normal modes harmonics excited by gravitational radiation. His finding was criticized due to a lack of more convincing statistical proof (Flinn, 1971). An important study was done by Dyson (1969) who was the first to calculate the response of a flat-Earth model to an incident GW, where eﬀects of sphericity and rotation were added to the flat stationary Earth solution. The calculated response was in the 1-Hz band where seismic wavelengths are small compared to the Earth’s radius and large compared to lateral density heterogeneities. Dyson (1969) showed that the GWs, in such a set-up, are absorbed only by irregularities in the shear-modulus, with the strongest absorption at free surface. De Sabbata et al. (1970) proposed detecting GWs by the observation of Earth’s free os-cillations. Their apparatus consisted of laser interferometer which allows to measure the soil deformations. They proposed that distinction of the seismic (free oscillations) from the gravitational signals could be accomplished by considering a long interval of time, to look at the Fourier components at the presumed frequencies and consider the decay time of the oscillations. Mast et al. (1974) performed the search for gravitational radiation from pulsars using a seismometer on the Earth. Even though no signal was found, they esti-mated an upper limit on the Earth motion due to such signal from 10−11 m near 1 Hz to 10−14 m near 125 Hz. Extensive work on the reception of GW by an elastic self-gravitating spherical detector was done by Ashby and Dreitlein (1975). The equations of motion of a detector are presented in the coordinate system of Fermi, where the GW field appears as a classical driving force, and exact analytic solutions are modeled for the homogeneous isotropic elastic sphere as well as self-stress sphere, where stress on the body due to its own gravitational field causes radial variations in density and elastic moduli in equilibrium state. The elastic response was calculated for monochromatic waves in the range 10−4 Hz to 1 Hz. Similar work was done by Linet (1984), where he modeled the equations governing the interaction between non-rotating elastic self-gravitating sphere and GWs. Based on Dyson (1969), Jensen (1979) analysed the absorption of GW in the 1-Hz band by the layered crust of a realistic Earth model, developing the interaction between GWs and the elastic continuum. Jensen (1979) showed that discontinuities in the elastic modulus in a layered model significantly enhance the response of Earth to GWs at specific frequencies. The com-plete response of the radially heterogeneous rotating and self-gravitating Earth in terms of induced toroidal and spheroidal motions was then developed by Ben-Menahem (1983). He showed that in the long-wavelength regime for the induced spheroidal vibration the most significant response corresponds to quadrupole modes. More recently, Coughlin and Harms (2014a,b,c) revisited Dyson’s and Ben-Menahem’s formalism of the Earth response to incident GW for the calculation of the upper limit of GW energy density. In the first paper (Coughlin and Harms, 2014a) they used a global network of broadband seismometers and they considered isotropic stochastic GW background integrated over one year in the frequency range 0.05-1 Hz. In the second paper (Coughlin and Harms, 2014b) they used data from a superconducting gravimeter network in the frequency range 0.035-0.15 Hz. In the third paper (Coughlin and Harms, 2014c) they used Apollo-era seismic data integrated over one year in the frequency range 0.1-1 Hz.
Besides Weber and Dyson, who were pioneers in considering the Earth as a detector of GW, many papers that followed studied the interaction of GW and elastic solids in the general relativity context. One of the first studies modeled a concept of the perfectly elastic solid in the high-pressure elasticity theory (condition that occurs in the interiors of neutron stars) for the purpose of scrutinizing the interaction of gravitational radiation with planetary bodies (Carter and Quintana, 1972). Also, for the fist time the strain-curvature equation for an elastic test body interacting with a GW was formulated in general relativistic systems (Glass and Winicourt, 1972). Further, the interaction problem was also solved in the gravito-inertial system of reference (Dozmorov, 1976a,b). In the later paper (Dozmorov, 1976b), it was emphasized the existence of the superposition of two diﬀerent elastic waves, those with the phase velocity equal to the speed of light and those with the phase velocity equal to the seismic velocity.

READ Classification and Localization Predictions with Deformable Parts

Spherical Non-rotating Elastic Isotropic Earth (SNREI)

A SNREI Earth model stands for the spherically symmetric, non-rotating, perfectly elastic and isotropic model. Isotropic in this case means that the initial stress is isotropic, so there is no deviatoric stress τ = 0, and that the fourth-order elastic tensor is isotropic, defined as ijkl = κ − 2 µ δij δkl + µ(δikδjl + δilδjk), (3.1).
where κ stands for isentropic incompressibility or bulk modulus and µ for rigidity or shear modulus. Incompressibility and rigidity may be specified as radial variations of the compressional-wave speed α = (κ + 43 )/ρ and shear-wave speed β = µ/ρ, where ρ is mass density. The most well known and used SNREI 1D Earth model is the Preliminary Reference Earth Model, henceforth referred as PREM (Dziewonski and Anderson, 1981). The model is polynomial in nature, hence it provides formulas for the seismic velocities, α and β, density and quality factors Q as a function of radius for various regions of the Earth. The original version is transversely isotropic (only 220 km in the outer mantle) as well as anelastic.
The linearized equations and boundary conditions governing the free oscillations of a SNREI Earth model can be obtained for a non-rotating, hydrostatic model (τ 0 = 0). The frequency-domain equation in terms of the displacement s and the incremental gravitational potential φ is defined as −ω2ρs − κ + 1 µ ∇(∇ · s) − µ∇2s − 3 1 −2∂r µ ∂rs + 2 ˆr × (∇ × s) ∂r κ − 2 ∂rµ (∇ · s)ˆr 3 2 sr )ˆr + ρ∇φ (3.2) + (4πGρ +ρg ∇sr − (∇ · s + 2r−1sr)ˆr = 0.
where the vector Laplacian is ∇2s = ∇(∇ · s) − ∇ × (∇ × s). The kinematic boundary conditions require the displacement to be continuous everywhere except on the fluid-solid boundaries, where tangential slip is allowed. The dynamical boundaries are ˆr · T = 0 on ∂⊗, (3.3).

Rotating Anelastic Heterogeneous Earth

In most global seismological applications eﬀects of Earth’s rotation, hydrostatic ellipticity and lateral heterogeneity can be regarded as slight perturbations from the equilibrium state. Usually in those applications, the normal-mode perturbation theory is used to calculate the singlet eigenfrequencies and associated eigenfunctions of the perturbed Earth. The basic problem to start with is finding the perturbation to a non-degenerate eigenfrequency of a mode that is well isolated in the seismic spectrum. In the first-order perturbation theory one wants to find the eigenfrequency perturbation δω without solving simultaneously for the perturbations in the associated eigenfunctions δs. Solutions to this classical problem serves as a basis for the degenerate and quasi-degenerate perturbations problem, where we cannot treat modes as perfectly isolated in the normal-mode spectrum. As mentioned before, the real eigenfrequencies of spherically non-rotating symmetric Earth model exist in (2l + 1) degenerate spheroidal and toroidal multiplets. This degeneracy is removed in a three-dimensional rotating Earth model and it is perceived as splitting of the multiplet eigenfrequencies and coupling between singlets within the multiplet and also between indi-vidual multiplets, if their unperturbed eigenfrequencies are in close proximity to each other for later case. In the splitting and coupling approaches the basis functions are the unper-turbed multiplet eigenfunctions of the mode one wishes to investigate, and the perturbed singlet eigenfunctions are of the form s = k qksk, where qk are expansion coeﬃcients to be determined and sk are the singlet eigenfunctions on a SNREI Earth model. Theoret-ically, all eigenfunctions form the basis set, however since it is impossible to incorporate n → ∞ modes into the computation, one needs to truncate the number of studied multi-plets, hence the term quasi-degenerate multiplets. The split eigenfrequencies are treated as small perturbation away from positive reference or fiducial frequency ω0 .

The Splitting of an Isolated Multiplet

In the isolated-multiplet approximation the splitting of and self-coupling between singlets within the target multiplet, due to the rotation, ellipticity and lateral heterogeneity, is governed by a splitting matrix defined as H=W+(2ω )−1 Vell+cen + Vlat + iA − ω2 (Tell + Tlat) , (3.14).
where W is rotating matrix, which contains the first-order perturbations of the Earth’s rotation without the centrifugal potential and without associated ellipticity perturbations. Further, the combined eﬀects of rotation and hydrostatic ellipticity are represented by matrices (2ω0)−1(Vell+cen − ω02Tell) with Tell being the kinetic-energy matrix and Vell+cen the elliptical-plus-centrifugal potential energy matrix. Non-hydrostatic lateral heterogeneity perturbations are contained in (2ω0)−1(Vlat − ω02Tlat) matrices and A is the matrix of anelastic perturbations. Therefore, the matrices W, Vell+cen, Vlat, Tell and Tlat contain all deviations from the SNREI Earth model, so they are also called the perturbation matrices. The perturbation splitting matrix H is fundamentally an operator defining an ordinary eigenvalue problem for the complex eigenfrequency perturbations. In the isolated-multiplet approximation the size of the matrix is (2l + 1) × (2l + 1), therefore the coupling between adjacent multiplets is ignored. The elements of this complete (2l + 1) × (2l + 1) self-coupling matrix are given by Hmm′ = ω0[ibmδl−l′ + (a + cm2)δmm′ ] + ω0 (cst + iψst) YlmYstYlm′ dΩ.

Table of contents :

Acknowledgements
Extended Abstract
Résumé Étendu
List of Figures
List of Tables
1 Introduction
2 Detection of Gravitational Waves
2.1 Sources
2.2 Detectors and Frequency Ranges
2.3 Elastic Bodies as Detectors
3 Earth Normal Modes
3.1 Spherical Non-rotating Elastic Isotropic Earth (SNREI)
3.2 Rotating Anelastic Heterogeneous Earth
3.2.1 The Splitting of an Isolated Multiplet
3.2.2 The Mode Coupling
3.2.3 The Splitting Function Coefficients
3.3 Green tensor
4 Normal Modes Excited By Gravitational Waves
4.1 Force Term in Flat Space-Time Approximation
4.2 Terrestrial Reference System and Elastic, Non-rotating Earth ModeL
4.2.1 Metric Perturbation Defined as Plane-wave
4.2.2 Green tensor
4.2.3 Induced spheroidal response
4.2.4 Discussion
4.2.5 Conclusion
4.3 Celestial Reference System and Anelastic, Rotating Earth Model
4.3.1 Rotation matrix from Celestial to Terrestrial Reference System
4.3.2 Metric perturbation for the binary star system
4.3.3 Green tensor
4.3.4 Induced spheroidal response
4.3.5 Discussion
4.3.6 Conclusion
5 Search for Gravitational Waves Using Matched Filtering
5.1 Introduction
5.2 Synthetic tests
5.3 Observations
5.4 Conclusion
6 Uncertainties in Normal Mode Studies
6.1 Introduction
6.2 Synthetic Experiments
6.3 Observations
6.4 Conclusions
7 Conclusion and Perspectives
A Greenwich Sidereal Time
B Matched Filter And Detection Statistics
C Autoregressive Method in Frequency Domain
D Optimal Sequence Estimation
E Phasor Walkout
Bibliography