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Gravitational Waves: Generation and Detection Gravitational waves: Generation and Detection Joseph Bayley Supervisor: Prof. Stewart Boogert November 6, 2015 Abstract After the first indirect observation of gravitational waves from the Hulse-Taylor binary system in 1975, many groups have set out to try and directly detect them. These groups use a range of methods of detection, each of these detectors have advanced substantially since the initial work on the experimental detection of gravitational waves, carried out by Joseph Weber. One of the leading methods in the detection of gravitational waves is interferometry; the most recent upgrades to the laser interferometry gravitational observatory (LIGO) pro- vides a high probability of detection. The interferometric experiments operate in the high frequency range where the coalescence of binary neutron stars generate gravitational waves, the most recent upgrades are expected to be able to detect up to 40 of these coalescences per year. Other experiments such as pulsar timing arrays focus on a lower frequency range, where they hope to detect sources of the gravitational wave background from supermassive black holes. Each of the experiments sensitivity is determined by its signal to noise ratio, therefore a lot of the recent developments in the search for gravitational waves have been in the reduction of noise in the detectors. This review focusses on the generation and de- tection on gravitational waves, particularly on the interferometry experiments methods of noise reduction and results. 1 Contents 1 Introduction 3 1.1 Gravitational Waves and their properties . .3 2 Sources of Gravitational Waves 7 2.1 Bursts . .8 2.1.1 Binary mergers . .8 2.1.2 Core collapse . .9 2.2 Chirp signals . .9 2.3 Continuous signals . 10 2.4 Stochastic background . 11 2.4.1 Cosmological gravitational wave background . 11 2.4.2 Astrophysical gravitational wave background . 12 3 Detectors 12 3.1 Observations of the cosmic microwave background . 13 3.2 Pulsar Timing Arrays . 14 3.3 Resonant Mass Detectors . 14 3.4 Interferometers . 18 3.4.1 Shot noise . 19 3.4.2 Thermal noise . 23 3.4.3 Seismic noise . 23 3.4.4 Results . 25 4 Future searches for GW 29 5 Conclusion 31 2 1 Introduction Gravitational waves (GW) were first predicted by Einstein as a consequence of his theory of general relativity and were theorised to transfer energy in the form of gravitational radiation [1]. Since this prediction many groups have attempted to detect GW. Despite the fact that to date no signals have been detected, the recent upgrades to the laser interferometry gravitational observatory (LIGO), and advances in other detectors provide a promising outlook for their de- tection. Other than for the discovery itself, there are many motivations for finding GWs. Due to gravity being such a weak force, the GW are not absorbed by clouds of interstellar dust like electromagnetic radiation (EMR), this allows an unobstructed view of the universe. Other motivations for finding GW include the fact that they are emitted by some of the most ener- getic events in the universe. These events include the formation of black holes from compact binary coalescences and core collapse supernova (CCSN). GW from these sources would allow the study of black hole progenitors and their formation. As well as the observational benefits that GW offer, they also provide a way to test general relativity itself. After the prediction of GW from Einstein, it was not until 1975 that physical evidence of GW were found [2]. The first indirect observation of GW was from the famous Hulse-Taylor binary system, or pulsar B1913+16, which was the first binary pulsar system to be discovered [2]. This binary orbit was studied for 30 years and the radius of its orbit was found to decrease with time, this means the system must be losing energy and can be accounted for somewhere. The loss in energy from the orbit matched incredibly well with the prediction from general relativity, therefore the loss in energy is assumed to be from the emission of GW. Figure 1 from Ref. [2] shows how well the prediction matched with the observation. The result from the study of the Hulse-Taylor binary system gave a promising outlook for the existence of gravitational waves and led to the construction of many of the experiments that exist today. Although GW have not been directly detected, many groups and experiments are currently looking to do this. These experiments have to be incredibly sensitive; for example if two stellar mass black holes were close to coalescence at a distance of 1 Mpc, the LIGO experiment will have to detect a strain of ∼ 10−13 m across its 4 km arms [3]. There are a number of different methods that are used to try to detect the displacement caused by a GW. The first experiment was a resonant mass bar designed by Joseph Weber [4], this uses a solid aluminium bar with a particular resonant frequency. If a passing GW is of a similar frequency then it will cause the bar to resonate, this resonance can then be measured. Although many of the bars have been decommissioned there are more recent resonant mass experiments such as MINIGrail [5] which use spherical masses for detection. However the resonant masses have a limited sensitivity, therefore a different type of experiment was developed using interferometry. The largest interferometric experiment is the laser interferometry gravitational observatory (LIGO) [6], this is expected to be the first to directly detect GW, especially after its most recent upgrade. Other more indirect methods include pulsar timing arrays (PTAs), which exploit the regularity of millisecond pulsars [7], and experiments that look at patterns within the cosmic microwave background (CMB). Future detectors that are planned include the laser interferometry space antenna (LISA) [8], this uses a similar technique to LIGO however operates over a distance of 5×106 km [8] in space. The future detectors therefore hope to be able to reduce their sensitivity to a level that allows for the direct detection of a GW. 1.1 Gravitational Waves and their properties In Einstein's theory of general relativity, gravity is caused by distortions in spacetime, which are created by a mass or energy. The larger the mass of an object is, the more it will distort 3 General Relativity prediction Cumulative shift of periastron/s shift time Cumulative Year Figure 1: This plot shows the shift in periastron time, which is the change in time of when the stars are closet together, plotted against the time it was observed. It shows how the orbit of pulsar B1913+16 decayed over the course of 30 years and how well it matched the general relativity prediction [2]. 4 spacetime. Figure. 2 shows how spacetime is distorted by three different sized masses. Infor- Figure 2: This is an illustration of how three different masses distort spacetime. The smallest mass on the left to the largest on the right [9]. mation about the curvature of spacetime that these masses create is contained in a parameter called the metric tensor, gµν . The relationship between matter and the curvature of spacetime is contained within Einstein's field equation, this is written as 1 8πG R − Rg + Λg = T ; (1) µν 2 µν µν c4 µν where Rµν is the Ricci curvature tensor, R is the scalar curvature, gµν is the metric tensor, Λ is the cosmological constant, G is the gravitational constant, c is the speed of light and Tµν is the stress-energy tensor. By expanding Einstein's equations around flat space time ηµν, it is possible to find the wave equation for gravitational waves. For flat space-time 2−1 0 0 03 6 0 1 0 07 ηµν = 6 7 ; (2) 4 0 0 1 05 0 0 0 1 where ηµν is the Minkowski metric tensor in the coordinate system of (t; x; y; z). By perturbing the metric tensor gµν by hµν, it can be written as gµν = ηµν + hµν: (3) The expansion of the equations of motion to linear order hµν is then called linearised theory [1]. The linearised equations of motion can be written more simply by defining 1 h¯ = h + η h; (4) µν µν 2 µν µν where h = η hµν [1]. By adopting the Lorentz gauge in which ν @ h¯µν = 0; (5) the linearisation of Einstein's equation can be written as a set of wave equations 16πG 2h¯ = − T ; (6) µν c4 µν 2 2 2 where 2 = −(1=c )@t + 5 [1]. This shows that the gravitational radiation takes energy away from the source in the form of 5 a wave, distorting space time as it travels. Equation 6 shows the result of generating a GW in linearised theory, however to see how a GW interacts with test masses, its behaviour outside of the source needs to be found. In Eq. 6, the stress-energy tensor Tµν is equal to zero outside of the source[1], therefore Eq. 6 becomes 1 2h¯ = 52h¯ − h¯ = 0: (7) µν µν c2 µν Any wave in three dimensional cartesian coordinates satisfies the equation @2u @2u @2u @2u 1 @2u + + − = 52u − = 2u = 0; (8) @x2 @y2 @z2 @t2 v2 @t2 where v is the velocity of the wave and u(t; x; y; z) is the wavefunction [10]. By comparing this to the gravitational wave equation Eq. 7, it is clear that gravitational waves propagate at the speed of light c. To solve Eq. 7 another gauge is adopted called the transverse-traceless (TT) gauge. In this gauge only the spatial components are non zero as hµ0 = 0, and they are transverse to the direction of propagation [1].
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