In describing GWs, it is useful to consider how they would
act on a ring of free particles. In the figure below, consider
an intially circular ring of massive particles being acted upon by GWs
approaching perpendicular to the plane of the ring. The ring
is now distorted and oscillates between two ellipses separated by a
90 degree rotation. A GW has two possible polarizations called "+"
(pronounced "plus") amd "x" (pronounced "cross"). The ellipses in
these two different polarization states are rotated by 45 degrees.
All other GWs are some juxtaposition of these two states.
This diagram illustrates the effect of a GW on a ring
of free particles.
Now, imagine that our ring of particles is itself a sort of GW detector. That is, it is responding to GWs that originated from some other source. So consider, for example, the + polarization: the test particles are stretched right and left as they are simultaneously being squeezed top and bottom. Then they are stretched top to bottom as they are squeezed right and left, and so on. Spacetime is distorted during this process and the distance between points in spacetime changes, depending on the polarization of the wave. Thus, by measuring the relative motions of the particles, we can determine the polarization of the GWs acting upon them. And accordingly, if we know the polarization of a GW, we can make direct inferences about the properties of the source.
The stronger the GW, the greater the relative length change, or strain, that will occur between two points in spacetime. The strain, h, essentially measures the strength of a GW and is defined:
h = delta L / L
The amplitude of a GW, h(g), is defined as twice the strain.
The amplitude of GWs emitted by a massive object is proportional to the
second time derivative of its quadrupole moment, Q''. Spherically
symmetrical objects have Q''= 0--and thus do not emit GWs--while asymmetrical
objects have some Q''> 0. Massive objects with high acceleration
yield high Q'' which, in turn, yields GWs with large h(g). The target
sensitivity for first generation ground-based detectors, such as LIGO,
is h(g) ~ 10^-21.
Finally, it might be useful for some to compare gravitational
and EM radiation. The following table enumerates some of the differences.
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spacetime itself |
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Detectable EM radiation usually originates from the surface of objects, whereas GWs can tell us about the inside of an object, too, since they are the immediate results of a system's mass distribution. Thus, astronomers expect that the universe as seen in GWs will look very different from the universe seen in EM radiation.
Hulse and Taylor recorded the observed time of periastron (the minimum separation of an object to the center of mass of a binary system in elliptical orbits) for the pulsar over a number of years. In 1983, they reported that there was a systematic shifting in periastron time--the pulse signal was arriving sooner and sooner than before. In 1982, PSR 1913+16 was arriving at its periastron over a second earlier than it had been in 1974. That is, it was arriving over a second earlier than what would have been expected had its orbit remained constant since 1974.
Hulse and Taylor found that shifts in periastron time were consistent with the GR prediction that, with time, such a binary system will lose energy to gravitational radiation, resulting in a shorter orbital period.
The arrival time of PSR 1913+16 shortened, and
constinues to shorten, over the years, averaging a loss of about 76 millionths
of a second per year.
It is expect to continue this trend and coalesce with
its companion star approximately 300 million years from now.
The orbit of PSR 1913+16 shrinks at a rate of approximately 3 mm per orbit. At this rate, the two stars should merge in around 300 million years. It is hoped that instruments such as LIGO (the Laser Interferometer Gravitational-Wave Observatory) and LISA (Laser Interferometer Space Antenna) will be able to detect the gravitational radiation produced during such events.
How it works: LIGO, the joint project of Caltech and MIT, is an interferometer designed to detect the tiny distortions in spacetime resulting from the sudden motion of large masses. The range of frequencies that this ground-based observatory should be able to detect is 40 Hz to 1000 Hz. LIGO consists of three L-shaped detectors--two residing in Hanford, Washington and the other, in Livingston, Louisiana. The detectors have antennae that are 4 km long and perpendicular to each other, so as to be aligned with the polarization of incoming GWs. (The second detector at Hanford has 2 km arms and shares the same space as the other.) That is, one of the antenna will be able to detect the "squeezing" of space, while the other detects the "stretching." Laser light, because it is coherent and has a precise wavelength, is used to measure this squeezing and stretching. The laser is split into two beams--one beam propagating through each antenna, and reflected off of mirrors (the test masses) suspended in vacuum tanks before recombining at their vertex. The detection of GWs occurs when the well-known (undisturbed) path length of a beam is strained. Because the sensitivity of the detector increases the longer the arms, (and since it is impractical and expensive to extend the arms over distances much greater than 4 km), each laser beam is bounced between mirrors 50 times, thereby increasing its path length, before recombining at the photodetector. In order to confirm that a detection is the result of an actual cosmological event and not an anomalous signal, the outputs from the two observatories will be compared for events occuring within 10 milliseconds of each other--(0.01 light-seconds is the distance between Hanford, WA and Livingston, LA).
Hanford, Washington
Livingston, Louisiana
diagram of LIGO's antennae
Noise: The primary sources of noise for LIGO are seismic noise, thermal noise, and shot noise. Below, these noise sources, and the efforts employed to combat them, are discussed.
Looking at the above figure of LIGO's predicted noise spectrum, and based on the discussion of its primary noise sources, LIGO is expected to have an open window between 40 Hz and 1000 Hz where it can detect GWs with amplitude h(g) ~ 10^-21 and above.
Other ground-based detectors, this generation and next: Other ground-based observatories currently being built or planned include France and Italy's VIRGO, Germany and Britain's GEO, Japan's TAMA, and Australia's AIGO. Recall, the target sensitivity for ground-based detectors is currently h(g)~10^-21. And though instrumental experiments with first-generation LIGO have yet to be completed, detailed plans for increasing its sensitivity is underway. It is the goal of second- and third-generation LIGO to systematically increase the target sensitivity so that legitimate GW detections will be assured by the end of this decade.
LISA
LISA, the Laser Interferometer Space Antenna, is an ESA-NASA project expected to launch in 2011. This space-based antenna, consisting of three identical independently floating satellites arranged in an equilateral, will be designed to detect low-frequency GWs, ranging from 0.1 mHz to 1 Hz. Lisa's orbit around the Sun will lag Earth's by 20 degrees, with its orbital plane tilted 60 degrees to the ecliptic. The main two advantages of LISA over LIGO are obvious: First, LISA's lower bound on frequencies will not be limited by seismic noise. Second, the 5 million km sides of the triangle will allow the interferometer make measurements over 100 times more sensative than ground-based detectors.
How it will work: LISA will work in essentially the same manner as LIGO--a GW will cause a squeezing or stretching of space, thereby changing the pathlength of the laser beams reflecting between the test masses. However, LISA will not use beam-splitters; each satellite will emit two independent beams to the others. Furthermore, since a laser beam will be weakened after it has traveled such a long distance, and would weaken even more if it were to be reflected, the satellite to which it is sent emits a new laser, in phase with the former, upon being triggered.
LISA's configuration: The interferometer will consist
of three identical spacecraft, whose orbital plane is tilted 60 degrees
to the ecliptic.
Noise and other problems: LISA also departs from LIGO in the types noise expected to mask GWs.
LIGO vs. LISA: positional accuracy, signal-to-noise strength, and resolution
Once GWs are detected by an interferometer, the next step, of course, is to determine where they came from. Accurately determining the location of an object depends on one, or a combination, of the following conditions:
This plot compares the noise spectra of LIGO and LISA.
Each generation of LIGO is expected to improve the sensitivity of
the former by factors of ~10. The horizontal dashed
line indicates the expected background spectum of white dwarf binaries.
Massive, non-spherical systems are the prime candidates for GW detection. Below, I describe three of these sources,--merging binaries, supermassive black holes, and the stochastic background--the frequency bands in which they're expected to be observed, and our scientific goals for studying them. This list is not exhaustive--other candidates for GW detection include binary stars, supernovae with some inherent instabilities, and rotating or unstable neutron stars.
Merging compact binary systems
Such systems could consist of two neutron stars (NS), two black holes (BH), or one NS and one BH. Because these stars have huge masses and are close together, they rotate rapidly, thereby giving them a large Q''. They start out by spiraling each other at some distance. When their orbital separation is on the order of kilometers, they radiate an enormous amount of energy in a short time. The emitted GWs increase in amplitude and this waveform is called a chirp; (such systems are often called chirping binaries.) Eventually, the stars will coalesce. The GW waveform during coalescence is modeled now using very complicated computer simulations. It is hoped that we will be able to map this waveform from observations with ground-based detectors. When they have merged into a single star, the system de-excites by emitting GWs. The waveform during this phase is also well-known.
The coalescence of a compact binary system can be summarized
in three phases: (1) The waveform during the inspiral (red)
is well-known by astronomers. As the stars orbital
separation decreases, they release more energy in the form of GWs,
whose amplitude increases correspondingly. (2)
The waveform during the merging phase (blue) is still a mystery to
astronomers. (3) Once merged, the single star de-excites,
undergoing linear pulsations, and the emitted GWs decay in amplitude.
It is predicted that merging compact binary systems emit
GWs in the 10 Hz to 1000 Hz frequency bands. Thus, ground-based observatories
should be capable of detecting such events. Besides, perhaps, proving
the existence of GWs and confirming GR, such systems are of significance
because some are standard candles. We should be capable of determining
the distance to these objects by analyzing their waveforms.
Supermassive black holes
The detection of supermassive black hole (SMBH) events is one of LISA's primary goals. Since they emit little or no EM radiation during events, such as merging, GW detectors will be breaking completely new ground here, giving us insight into events never before observed directly, even by EM telescopes.
At present, one of the most exciting topics in astronomy
concerns the mystery of why the number density of AGN (active galactic
nuclei) decreases with increasing redshift (z). A popular hypothesis
is that SMBHs, (BHs with thousands to billions of solar masses) are at
the center of every galaxy. SMBH "events" include coalescences,
collapses of individual SMBHs, and gravitational slingshots. By measuring
such events we hope to determine SMBH masses, the mechanism by which they
form, their rate of formation, and the rate at which they coalesce or collapse.
Measuring these different population statistics would improve our understanding
of galaxy formation and the specific role of SMBHs in this process.
Stochastic background
Just as density fluctuations in
the early universe resulted in the anisotropic cosmic microwave background,
so might these initial perturbations have caused a stochastic background
of GWs. The most important difference between the stochastic background
and the cosmic microwave background is that, since they couple so weekly
to matter, GWs did not thermalize. Thus, the GW spectrum should come
to us unaltered from whatever produced it! If we were able to detect
the stochastic background, we would be able to make inferences about a
much earlier universe than we have been able to do with EM radiation.
But, though measuring the stochastic background is highly anticipated,
whether we will be able to observe it is very unpredicatable since the
sensitivity required to detect these fluctuations is far below the capablilities
of the present generation of GW detectors.
At approximately 10^-43 seconds, or 10^32 K, gravity separates from
the other three fources. The inflation period begins at about
10^-35 seconds.
The following table summarizes possible sources, the frequency
bands in which we expect to observe them, and the corresponding detection
methods.
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It is worth noting that GW observatories will also work in conjunction with optical observatories. We should be able to use GW detectors as "alarms" for EM telescopes. For instance, it is hypothesized that the coalescence of compact binary systems or the sudden collapse of supermassive objects into SMBHs are a source of gamma-ray bursts. Since we know the signature waveform of GWs in the stage just prior to coalescence or collapse, the detection of such signals would provide ample time to point EM telescopes in the direction of those events.
Ultimately, though careful predictions have been made about the range of GW frequencies and amplitudes we expect to observe from given sources, the outcome of GW astronomy is very unpredictable.
Page created by Nia Imara
UC Berkeley Astronomy, ASTR 228C
Presented Novermber 17, 2003