Gravitational Waves Gravitational wave astronomy Virtually everything you and I know about the Cosmos has been discovered via electromagnetic observations. Some inormation has recently been gleaned rom astro-particle observations (neutrinos and cosmic rays) We are now entering an era where we are using a wide variety o detectors to probe the Cosmos with gravity. A ull relativistic derivation o the existence and action o gravitational waves is beyond our scope here, but can be obtained by linearizing the ield equations o general relativity. Here let s convince our intuition. The idea that gravitational inormation can propagate is a consequence o special relativity: nothing can travel aster than the ultimate speed limit, c Imagine observing a distant binary star and trying to measure the gravitational ield at your location. It is the sum o the ield rom the two individual components o the binary, located at distances r 1 and r 2 rom you. As the binary evolves in its orbit, the masses change their position with respect to you, and so the gravitational ield must change. It takes time or that inormation to propagate rom the binary to you t propagate = d/c, where d is the luminosity distance to the binary. The propagating eect o that inormation is known as gravitational radiation, which you should think o in analogy with the perhaps more amiliar electromagnetic radiation Far rom a source (like the aorementioned binary) we see the gravitational radiation ield oscillating and these propagating oscillating disturbances are called gravitational waves. Like electromagnetic waves Gravitational waves are characterized by a wavelength λ and a requency Gravitational waves travel at the speed o light, where c = λ Gravitational waves come in two polarization states (called + and ) 1 Astrophysics Lecture
Gravitational wave observatories I you want to detect gravitational waves, you have to know what their physical eect on matter is what do they do to things you can tape together and call a detector The undamental inluence o gravitational waves is to change the proper distance between points in spacetime. That means i you have two ree particles, and monitor the spacetime interval between them, a gravitational wave will change that interval as it passes. The dierent polarization states have dierent eects on an array o particles. Imagine a ring o ree test particles. The + and states are named by the distortions they produce on the ring. For a gravitational wave propagating into or out o the page, the eects are shown below or a + polarization (in A) and or a polarization (in B) A +y +x +y +y +x +x B C 2 Astrophysics Lecture
Gravitational wave detectors then are devices that enable you to measure these kinds o distortions. Since the 1960 s man-made detectors have ocused on three undamental technologies Bar detectors. Here the elements o the ring should be thought o as the atoms in a large aluminum cylinder (or, in the limiting case, the mass elements on the ends o the cylinder). Roughly sensitive to gravitational waves with requencies near the resonant requencies o the bar. Intererometers. Here the elements o the ring should be thought o as the end mirrors in a traditional Michelson intererometer. Roughly sensitive to gravitational waves with wavelengths comparable to the length o the intererometer arms. Astrophysical detectors. using astrophysical systems, such as pulsars and the Cosmic Microwave Background, to detect extremely low-requency gravitational waves. For the past 40 years, a world network o gravitational wave detectors has been growing up. Bar Network. The irst network consisted o resonant bar detectors around the world, including Niobe, Explorer, Nautilus, Auriga and Allegro. With the advent o broadband intererometric detectors, these instruments have all but shut down operations. Intererometer Network. Beginning in the late 1980s, a network o ground based intererometric observatories sprang up and is still growing. Currently it is comprised o two LIGO detectors in the United States, the VIRGO detector in Italy, and the GEO 3 Astrophysics Lecture
detector in Germany. The TAMA detector in Tokyo has been shut-down, but the Japanese are now ramping up to build a new detector called LCGT (Large Cryogenic Gravitational-wave Telescope). Space-based Intererometers. Space based gravitational wave detectors like LISA or OMEGA are still at least a decade away. Pulsar Timing/CMB Polarization. There is a growing eort to search or low requency gravitational waves. Most notably this includes the radio pulsar timing collaboration, NanoGrav. There is also a dedicated collaboration o microwave astronomers building telescopes in Chile (ACTPol) and the South Pole (SPT) capable o detecting gravitational wave imprints in the polarization o the cosmic microwave background. Detector Sensitivity I you build a detector, the principle goal is to determine what gravitational waves the instrument will be sensitive to. We characterize the noise in the instrument and the instrument s response to gravitational waves using a sensitivity curve. Sensitivity curves plot the strength a source must have, as a unction o gravitational wave requency, to be detectable. There are two standard curves used by the community: Strain Sensitivity. This plots the gravitational wave strain amplitude h versus gravitational wave requency. Strain Spectral Amplitude. This plots the square root o the power spectral density, h = S h versus gravitational wave requency. The power spectral density is the power per unit requency and is oten a more desirable quantity to work with because gravitational wave sources oten evolve dramatically in requency during observations. The sensitivity or LIGO and LISA are shown below. Your own LISA curves can be created using the online tool at www.srl.caltech.edu/~shane/sensitivity/makecurve.html. 4 Astrophysics Lecture
I you are talking about observing sources that are evolving slowly (the are approximately monochromatic) then the spectral amplitude and strain are related by h = h T obs I you are talking about a short-lived ( bursting ) source with a characteristic width τ, then to a good approximation the bandwidth o the source in requency space is τ 1 and the spectral amplitude and strain are related by h = h = h τ The strain sensitivity o a detector, h D, builds up over time. I you know the observation time T obs and the spectral amplitude curve (like those plotted above) you can convert between the two via h D = hd T obs Sensitivity curves are used to determine whether or not a source is detectable. Rudimentarily, i the strength o the source places it above the sensitivity curve, it can be detected! The undamental metric or detection is the SNR ρ (signal to noise ratio) deined as ρ hsrc h D To use this you need to know how to compute h src. 5 Astrophysics Lecture
Pocket Gravitational Waves: Source Basics Binaries are expected to be among the most prevalent o gravitational wave sources. It is useul to have a set o pocket ormulae or quickly estimating their characteristics. Consider a gravitational wave binary with masses m 1 and m 2, in a circular orbit with gravitational wave requency = 2 orb, then: chirp mass M c = (m 1m 2 ) 3/5 (m 1 + m 2 ) 1/5 scaling amplitude chirp h o = G c 2 M c D = 96 c 3 5 G M c ( ) 2/3 G c 3πM c ( G c 3πM c ) 8/3 The chirp indicates that as gravitational waves are emitted, they carry energy away rom the binary. The gravitational binding energy decreases, and the orbital requency increases. The phase φ(t) o the orbit evolves in time as ( φ(t) = 2π t + 1 ) 2 t 2 + φ o, where is the chirp given above, and φ o is the initial orbital phase o the binary. Luminosity Distance rom Chirping Binaries Suppose I can measure the chirp and the gravitational wave amplitude h o. The chirp can be inverted to give the chirp mass: [ ] 3/5 M c = c3 5 G 96 π 8/3 11/3 I this chirp mass is used in the amplitude equation, one can solve or the luminosity distance D: D = 5 c 96π 2 h o 3 This is a method o measuring the luminosity distance using only gravitational wave observables! This is extremely useul as an independent distance indicator in astronomy. Working with chirping binaries Chirping binaries are neither monochromatic, nor are they bursting. They are leisurely evolving through the observing band. How can they be plotted on a sensitivity curve? 6 Astrophysics Lecture
First we must estimate what chirping is even detectable. We can use the chirp = d/dt to estimate the time it takes a circularized binary to evolve between any two requencies 1 and 2 : [ ] t = κ 8/3 1 8/3 2 where κ = 5 ( ) 5/3 G 256 c M 3 c π 8/3 The requency resolution o the detector is δ = 1/T obs. A chirp can be detected i the requency 2 at the end o the observation is 2 1 + δ The requency at which a source spends t = T obs crossing δ is called the stationary requency, s. To ind the stationary requency, set: t = T obs 2 = 1 + δ = 1 (1 + δ/ 1 ) then [ ] t = κ 8/3 1 8/3 2 T obs = κ 8/3 1 [ ] T obs = κ 8/3 1 8/3 1 (1 + δ/ 1 ) 8/3 [ ( [ 1 (1 + δ/1 ) 8/3] κ 8/3 1 1 1 8 )] δ 3 1 T obs κ 8/3 1 [ 8 3 ] [ δ 8 = κ 11/3 3 1 1 T obs ] [ 8 1 = s 3 κ T 2 obs ] 3/11 For all sources with requency s, they appear monochromatic over the observing time. I s, then the time it spends between 1 and 2 = 1 + δ (the time in a requency bin, t bin ) is given by t = T obs δ = 1/T obs 2 = 1 + δ where we have assumed δ t bin = 8 3 11/3 1 κ T obs 7 Astrophysics Lecture
At a given requency, the number o cycles N cy o gravitational radiation that a source emits depends on the time t bin that a source spends in the requency bin around that requency N cy = t bin 8 κ 8/3 3 T obs Strength o Gravitational Wave Binaries We have all the tools in hand at this point to make some basic estimates o source strength or binaries. At this stage, our goal is to determine the strain h src that could be plotted on a strain spectral amplitude sensitivity curve (h D vs. ) There are three basic cases we can address at this point: Monochromatic binaries. This are binaries or which s, so they do not chirp appreciably during an observation o length T obs. In this case the strength is h mono = h o T obs Chirping binaries. These sources have s and evolve across many requencies during an observation time. For these sources, the strain must be calculated and plotted at each requency! The value to plot depends on the number o cycles N cy emitted in a give requency bin, so h chirp = h o t bin Bursting sources. For short events with time proiles o width τ b and maximum burst amplitude h b the central requency o the burst is 1/τ b. For these sources, the strength is given by h burst = h b τ b 8 Astrophysics Lecture