CHAPTER 2 GRAVITATIONAL WAVES, SOURCES AND DETECTORS
Einstein’s theory of general relativity postulates that rapidly moving objects will emit energy. This energy produces periodic ripples or deformations in the fabric of space-time. These ripples are known as gravitational waves (GWs). GWs can be described by oscillations in the so-called fabric of space-time, causing space and everything in it to stretch and squeeze as the waves pass by. Propagating at the speed of light, GWs do not travel through the space-time, rather the fabric of space-time itself is oscillating, i.e. GWs can be thought as “messengers”. An important feature of GWs is that all astronomical bodies are transparent to GWs as they pass relatively unaffected through the matter. The GWs carry information about the sources at which they were produced such as information about the movements of stellar bodies, the structures and sizes of merging neutron stars or black-hole binaries and the map of space-time around super massive black holes.
However, the events producing GWs are at great distances from us. Even if caused by the most violent events in a nearby galaxy the GWs become very weak when they reach our planet. This makes the detection of GWs an extremely difficult task
GRAVITATIONAL WAVES, SOURCES AND DETECTORS
Until now any attempts to directly detect GWs have not been successful, however, astrophysicists are confident that GWs exist since there is a very strong evidence for their existence based on the indirect measurements of the orbital decay of binary neutron star system PSR1913+16, that are very consistent with the theoretical predictions of energy loss because of gravitational radiation [13, 14]
The Effect of GWs
The effect of a GW is to stretch and squeeze the space-time fabric in di-rections perpendicular to the direction of wave propagation. This can be observed by comparing the distance between two (or more) independent freely falling/floating test masses. The GWs are quadrupole, that is they occur in two fundamental states of polarization namely plus- (‘+’) and cross- (‘×’) polarization. The effect of GWs on a ring of free particles is shown in Figure 2.1. Here the GW is traveling perpendicular to the plane of the ring. The top panel shows the plus polarization while the bottom panel shows the cross polarization. The ring is distorted by the passing GWs and the effect is shown at different phases. Plus polarization changes the distance of free particles by first squeezing the ring along a horizontal direction and then along a vertical direction. Cross polarization has ba-sically the same effect but at 45◦ inclination. The total area of the plane remains the same and if the GWs are traveling along the plane then there will be no distortion at all.
As shown above the effect of GWs on free falling particles is that the distance between these particles changes as the wave passes. The change in the distance, called strain amplitude and traditionally denoted by h, is used for measuring the effect of a passing GW directly. However, this change in the distance is extremely small and requires highly sensitive detectors. There are two types of detectors used for the detection of GWs. One is a resonant mass detector in which large masses are used and the deformation caused by GWs in them is measured [15, 16]. The other type is a laser interferometric detector which actually works in a network of more than one detectors located at sufficiently long distance from each other [17, 18]. As the name indicates, laser interferometric detectors use laser interferometry to measure the changes in the strain amplitudes. When a GW passes through the plane of the detector, the distance between the masses changes by an amount L, where L is the distance between the two masses, known as arm-length, resulting in a strain amplitude h = L/L. There are several detectors around the world that use laser interferometry to detect GWs. The ground based detectors are GEO600 (Germany) , LIGO (Hanford, USA) and LIGO (Livingston, USA) , VIRGO (Italy) , and TAMA300 (Japan) . These ground based detectors are large L-shaped instruments with up to 4km long arms at 90◦ to each other. From the central station laser beams are sent to the ends of the arms where they are reflected by mirrors that are suspended by wires to work as approximately free-falling masses. When a GW passes through the plane of a detector the distance between the mirrors changes by a small amount, which is monitored by photo detectors that measure the phase change of the light. These detectors are at most sensitive to gravitational radiation in the range 1 − 104Hz.
A space-borne detector called Laser Interferometer Space Antenna (LISA) is currently being designed under a joint mission of NASA and ESA. The design of LISA allows to have very long arms and therefore is expected to be sensitive to low frequency gravitational radiation in the range 5 × 10−5 − 10−1Hz. The basic detector consists of three freely flying spacecraft located at the vertices of an imaginary equilateral triangle configuration with L = 5 × 10 9m long sides. Each spacecraft will carry two free-falling test masses and laser instruments that exchange laser beams with other two spacecraft to track the distances between the test masses within them to indicate the passage of a GW. The LISA triangle will move around the sun 20◦ ( ∼ 5.2 × 107km) behind the Earth with its guiding centre (a point which is equidistant from the three vertices) at the Earth’s orbit about the Sun. The LISA constellation is depicted in Figure 2.2.
LISA is not a pointed instrument, and can never be, rather it is an all-sky monitor and at any one time, LISA maps the whole sky. At different po-sitions it will have different sensitivities to GWs from a particular source depending on the location of the source and the polarisation of the waves. LISA will measure simultaneously both polarization components of the incoming GWs. The data will initially consist of at least three time series, recorded along three arms of the detector, from which all physical param-eters of the source, including its position, can be extracted. Mapping the whole sky means that there will be GWs from tens of thousands of resolv-able as well as unresolvable sources all overlapping in frequencies and phases [23, 24].
LISA will be observing signals from several different GW sources. Some of the sources which fall in the LISA sensitivity band are briefly mentioned as following
These types of binaries develop when two objects with very dense masses such as neutron stars (NSs) or white dwarfs (WDs), with roughly equal masses, orbit about each other. The compact objects move initially at an elliptical orbit about the common centre of mass. Over the course of time their individual orbits become increasingly circular as the two objects come closer and closer and at some point in time the two bodies appear to move around the centre of mass at the same circular orbit. This orbit decays with time like a spiral, as the two bodies come closer and closer to each other, which proceeds towards the common centre of the masses. Such spirals are called inspirals (in-spirals). As the two masses gradually inspiral, they emit gravitational radiation with a frequency falling well inside the LISA sensitivity band. These signals encode the luminosity dis-tance to a binary, its sky location, and information about other physical parameters. During its operation the LISA is expected to observe several thousands individual galactic binary systems [25, 26]
Mergers of Massive Black Hole Binaries
Similar to galactic binary systems are massive black hole binary systems in which both members are massive black holes with a total mass in range 105 M ⊙ − 109 M⊙. These sources will be detectable by LISA at extremely large distances due to large masses involved. The GWs from these sources encode information about the masses and the spins of the two members. Once such sources are detected and the physical parameters are estimated, this information can be used to find out how these massive black holes are formed and what is the rate of their mergers, that is how often massive black holes encounter each other, which will put light on the evolution and structure of the large scale galactic dynamics [27, 28].
Extreme Mass Ratio Inspirals
It is predicted that most galaxies, in their centres, host super massive black holes (∼ 105 M⊙ − 107 M⊙) surrounded by a dense population of stellar objects such as NSs, WDs or small black holes and other normal stars. Due to multi-body interactions these stellar objects are often pushed into an or-bit which passes too close to the central mass. The captured object then spirals in by orbital decay through the emission of gravitational radiation and eventually plunges into the central mass. Due to large differences between the two masses, such inspirals are called extreme mass ratio in-spirals (EMRIs), and are one of the most exciting sources to be observed by LISA. The EMRI studies will help to understand the structure of the space-time around the SMBH using the physical parameters such as spin and mass, and the interactions between SMBH and the cluster of stellar masses around it . The detection of these signals and the estimation of their physical parameters is the focus of this work and will be discussed in details in Chapter 3.
Cosmological Background, Stochastic Sources and Bursts
Stochastic GWs are random signals generated by large number of inde-pendent, incoherent and unresolved or diffuse sources. These signals can originate either from the early cosmological events or the astrophysical events happening throughout the history of the universe. The cosmologi-cal background could consist of the left over GWs that were produced as the result of the processes that took place very shortly after the big bang. The astrophysical background on the other hand is produced by very recent processes, such as supernova bursts and signals from millions of unresolved compact or massive black hole inspirals [30, 31]. These GWs might encode information about the early structure of the universe and other high energy astrophysical events. The superposition of GWs from these random and unresolved sources will form a very strong background noise which will be there in LISA data stream and therefore will be of great importance for the precise detection of the other sources
1.2 About this Work
1.3 Organization of this Thesis
2 Gravitational Waves, Sources and Detectors
2.1 Gravitational Waves
2.2 The Effect of GWs
2.3 GWs’ Detectors
2.4 LISA Sources
2.5 Data Analysis
3 EMRIs, Source Model and LISA Response
3.2 The AKWModel
3.3 Parameter Ranges
3.4 LISA Response
4.1 The Bayesian Approach
4.2 Monte Carlo Integration
4.3 Convergence Acceleration
4.4 Digital Signal Processing Methods
4.5 Bayesian Spectrum Analysis
4.6 Signal-to-Noise Ratio
5 Applications and Results
5.3 The Experimental Setup
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