DISSERTATION Search for cosmological gravitational-wave background at high frequencies Graduate School of Human and Environmental Studies, Kyoto University NISHIZAWA, Atsushi 28 November, 2008: ver.1.0 (22 July, 2009: updated ver.1.1) Abstract We describe the general framework for a cosmological gravitational-wave background (GWB) search with a laser-interferometric gravitational-wave (GW) detector. What we studied in this thesis can be divided into two topics: (i) the search for non-tensorial polarization modes (scalar- and vector-like polarizations) of a GWB with a large-scale laser-interferometric detector pair on the ground, (ii) the GWB search at ultra-high frequencies (» 100 MHz) with a pair of meter-sized laser-interferometric detectors. The ¯rst topic is involved in the theories with extra dimensions and the modi¯ed gravity theories. In the general relativity, there are two polarization modes of a GW. However, in the general theories of gravity, six polarization modes are allowed. If the extra polarization modes of GWB are detected, we can obtain some information about new physics. The search has not been performed at all so far with an interferometer. So, we extended the conventional formalism of a cross-correlation analysis to non-tensorial polarization modes and calculated the sensitivity to the GWB. We also discussed the detectability with real detector pairs. The second topic is the search for a GWB at ultra high frequencies. It is also im- portant because some models in cosmology and particle physics predict relatively large GWB at ultra-high frequency » 100 MHz. Upper limits on GWB in wide-frequency ranges have been obtained from various observations. However, they are all indirectly derived from the observations. As far as we know, little direct experiment has been done above 100 kHz except for a few experiments, though the direct constraint is much weaker than the constraints at other frequencies. Thus, a much tighter bound above 100 kHz is needed to test various theoretical models. First, we investigated the laser-interferometric detector designs that can e®ectively respond to GW at high frequencies, and found that the con¯guration, a so-called synchronous-recycling interferometer (SRI) is the best at these sensitivities. Then, we investigated the location and orientation dependence of two SRIs in detail, and derived the optimal location of the two detectors and the cross-correlation sensitivity to a GWB. We also describe the experiment done by our group and the results. These studies are not limited at the search at » 100 MHz, but can also be applied to the detectors in which the wavelength of a GW is comparable with the detector size. As a developed version of the SRI, we proposed a new detector design, a so-called resonant speed meter (RSM). The remarkable feature of this interferometer is that, at certain frequencies, gravitational-wave signals are ampli¯ed, while displacement noises are not. We also studied the quantum noise in a RSM, and its ultimate sensitivity. i Contents 1 Introduction 1 1.1 Gravitational waves . 1 1.2 Gravitational-wave sources . 1 1.3 Gravitational-wave detectors . 4 1.4 Outline and notation of the thesis . 8 2 Stochastic gravitational-wave background 11 2.1 Gravitational waves . 11 2.1.1 Linearized Einstein equation . 11 2.1.2 Gravitational waves . 12 2.1.3 Quadrupole nature of gravitational waves . 14 2.2 Stochastic gravitational-wave backgrounds . 14 2.2.1 Assumptions . 16 2.2.2 Energy density . 16 2.2.3 Characteristic amplitude . 17 2.2.4 Number density of gravitons . 18 2.2.5 Decoupling of gravitons . 18 2.2.6 Characteristic frequency . 19 3 Creation of cosmological GWB 21 3.1 Inflation . 21 3.2 GWB creation in de-Sitter inflation . 22 3.3 GWB creation in slow-roll inflation . 28 3.4 GWB creation in quintessential inflation . 32 3.4.1 Quintessential inflation . 32 3.4.2 GWB spectrum . 34 3.5 Pre-big-bang model . 36 3.6 Other production mechanisms . 39 4 Observational constraints on GWB 43 4.1 Big-bang nucleosynthesis limit . 43 4.2 CMB limit . 45 4.3 Pulsar-timing limit . 46 4.3.1 Limit from spinning pulsars . 47 4.3.2 Limit from binary pulsars . 49 iii CONTENTS iv 5 Direct search for GWB 51 5.1 Correlation analysis . 51 5.2 Overlap reduction functions . 54 5.2.1 Tensorial expansion . 54 5.2.2 Optimal con¯guration . 56 5.2.3 Overlap reduction functions of realistic detector pairs . 57 5.3 Observational constraints on GWB by LIGO . 60 5.4 Searching for non-tensorial polarizations of gravitational waves . 63 5.4.1 Non-tensorial polarization mode . 63 5.4.2 Angular response of a single detector . 65 5.4.3 Overlap reduction function . 68 5.4.4 Detectability . 72 6 Direct GWB search at ultra-high frequencies 77 6.1 Gravitational-wave sources at ultra-high frequencies . 78 6.2 Optimal detector design . 79 6.2.1 Interferometric-detector designs . 79 6.2.2 GW response functions . 81 6.2.3 Detector comparison . 84 6.3 Correlation of two detectors . 87 6.3.1 Identi¯cation of GW response and noise in a SRI . 87 6.3.2 Dependence of sensitivity on the relative locations between two detectors . 88 6.3.3 Sensitivity to GWB . 91 6.3.4 Cross-correlated noise . 93 6.4 Experimental search for GWB . 95 6.4.1 Experimental setup . 95 6.4.2 Data analysis . 97 7 Resonant speed meter 101 7.1 Detector design and sensitivity . 102 7.1.1 Detector response . 102 7.1.2 Noise curves . 106 7.2 Quantum noise in a RSM . 107 7.2.1 Input-output relation . 107 7.2.2 Spectral density . 112 7.3 Sensitivity to GWB . 114 8 Conclusions 117 8.1 Summary . 117 8.2 Discussions and future prospects . 118 A Bogolubov transformation and particle creation 121 B Formulae of spherical Bessel functions 123 v CONTENTS C The calculation of the overlap reduction function for non-tensorial modes 125 D GW polarizations in higher-dimensional spacetime 127 D.1 Pure 5-dimensional Minkowski spacetime . 127 D.2 Pure 6-dimensional Minkowski spacetime . 129 E GW response functions of interferometers: general expressions 131 E.1 Synchronous-recycling interferometer . 131 E.2 Fabry-Perot Michelson interferometer . 133 E.3 L-shaped cavity Michelson interferometer . 135 F Quantum theory in a laser interferometer 139 F.1 Quantum formalism . 139 F.2 Conventional Fabry-Perot Michelson interferometer . 140 F.3 Optional quantum con¯gurations . 143 Chapter 1 Introduction 1.1 Gravitational waves A gravitational wave (GW) is a ripple of spacetime, which propagates as a wave with the speed of light. The GW is predicted according to the theory of general relativity, published in 1916 by Einstein [1]. Nowadays, the many predictions of the General relativity have been con¯rmed in many observations and experiments [2]: gravitational lensing, Shapiro delay in the solar system, the perihelion advance of the planet Mercury, dragging of an inertial frame, etc.. In addition, the binary pulsars supply us with the splendid opportunities to test the general relativity, owing to the considerable stability and the strong gravity. From the observation of the change in the revolution period of B1931+16 (Hulse-Taylar binary pulsar), general relativity has been tested at a level of 1%, and the indirect evidence of the existence of GWs has been obtained [3, 4]. Recently, Valtonen et al. have claimed that a new indirect evidence is found by the observation of a binary system of two candidate black holes in the quasar OJ 287 [5]. These evidences lead us to strongly believe the existence of GWs. Nevertheless, GWs have not directly detected yet. Aiming for the ¯rst direct detection of GWs, many research groups have constructed large detectors and done observations with the sensitivity improved by degrees. 1.2 Gravitational-wave sources GWs are radiated by objects whose motion involves acceleration, provided that the motion is not spherically symmetric nor axisymmetric [6]. A number of GW sources have been theoretically predicted, as shown in Fig. 1.1 together with the sensitivities of GW detectors. Gravitational waves from astrophysical sources The promising GW sources with astronomical origins are violent events involving com- pact objects, such as supernovae, gamma-ray bursts, the binaries of neutron stars, black holes, and white dwarfs and their mergers, and spinning-down pulsars. Recent 1 1. Introduction 2 study and observational data suggested a part of supernovae and gamma-ray bursts are related to the mergers of neutron stars, black holes, and white dwarfs. The GWs from these systems are important in that not only do they provide the test in the strong regime of gravity, but also they will bring new information that one cannot obtain by the observations with electromagnetic waves. This is owing to the strong transparency of GWs. Electromagnetic waves are scattered at the outer layer of the stars and the plasma around it, and prevent us from directly seeing the core of the stars. In con- trast, GWs can escape from the dense regions of the stars and directly propagate to the Earth. Thus, we can investigate the equation of state in extremely high-density region such as an inner core of a neutron star, the strong regime of gravity and its environment around a black hole, the explosion mechanism of a supernova, the central engine of a gamma-ray burst, the inner region veiled by dust such as the accretion disk around a compact object, etc.. Figure 1.1: Schematic view of GW sources and detector sensitivities. In the ¯gure, the noise curves of LIGO, LCGT, LISA, and DECIGO are representationally shown.