List of Figures

View Image Figure 1:
In Einstein’s theory, gravitational waves have two independent polarizations. The effect on proper separations of particles in a circular ring in the (x,y)-plane due to a plus-polarized wave traveling in the z-direction is shown in (a) and due to a cross-polarized wave is shown in (b). The ring continuously gets deformed into one of the ellipses and back during the first half of a gravitational wave period and gets deformed into the other ellipse and back during the next half.
View Image Figure 2:
Mass-radius plot for gravitational wave sources. The horizontal axis is the total mass of a radiating system, and the vertical axis is its size. Typical values from various sources for ground-based and space-based detectors are shown. Lines give order-of-magnitude constraints and relations. Characteristic frequencies are estimated from f ∼ (G ρ∕4π )1∕2. The black-hole and binary lines are described in the text.
View Image Figure 3:
The relative orientation of the sky and detector frames (left panel) and the effect of a rotation by the angle ψ in the sky frame (left panel).
View Image Figure 4:
The antenna pattern of an interferometric detector (left panel) with the arms in the x-y plane and oriented along the two axes. The response F for waves coming from a certain direction is proportional to the distance to the point on the antenna pattern in that direction. Also shown is the fractional area in the sky over which the response exceeds a fraction 𝜖 of the maximum (right panel).
View Image Figure 5:
The right panel plots the noise amplitude spectrum, ∘ ------- f Sh(f), in three generations of ground-based interferometers. For the sake of clarity, we have only plotted initial and advanced LIGO and a possible third generation detector sensitivities. VIRGO has similar sensitivity to LIGO at the initial and advanced stages, and may surpass it at lower frequencies. Also shown are the expected amplitude spectrum of various narrow and broad-band astrophysical sources. The left panel is the same as the right except for the LISA detector. The SMBH sources are assumed to lie at a redshift of z = 1, but LISA can detect these sources with a good SNR practically anywhere in the universe. The curve labelled “Galactic WDBs” is the confusion background from the unresolvable Galactic population of white dwarf binaries.
View Image Figure 6:
The sensitivity of interferometers in terms of the limiting energy flux they can detect, Jy/Hz, (left panel) and in terms of the gravitational wave amplitude with lines of constant flux levels (right panel).
View Image Figure 7:
Time-frequency maps showing the track left by the inspiral of a small black hole falling into an SMBH as expected in LISA data. The left panel is for a central black hole without spin and the right panel is for a central black hole whose dimensionless spin parameter is q = 0.9.
View Image Figure 8:
Normal mode frequencies (left) and corresponding quality factors (right) of fundamental modes with l = 2,3,4, as a function of the dimensionless black hole spin j, for different values of m = l,...,0,...− l (for each l, different line styles from top to bottom correspond to decreasing values of m). Figure reprinted with permission from [78]. Ⓒ External LinkThe American Physical Society.
View Image Figure 9:
The smallest fraction of black hole mass in ringdown waveforms that is needed to observe the fundamental mode at a distance of 3 Gpc (left) for three values of the black hole spin, j = 0 (solid line) j = 0.80 (dashed line) and j = 0.98 (dot-dashed line) and the error in the measurement of the various parameters as a function of the black hole spin for the same mode (right). Figure reprinted with permission from [78]. Ⓒ External LinkThe American Physical Society.
View Image Figure 10:
Comparison of waveforms from the analytical EOB approach (left) and numerical relativity simulations (right) for the same initial conditions. The two approaches predict very similar values for the total energy emitted in gravitational waves and the final spin of the black hole. Figure from [96].
View Image Figure 11:
The final spin of a black hole that results from the merger of two equal mass black holes of aligned spins (top panel) and nonspinning unequal mass black holes (middle panel). The bottom panel shows the region in the parameter space that results in an overall flip in the spin-orbit orientation of the system. Figure reprinted with permission from [312]. See text for details. Ⓒ External LinkThe American Astronomical Society.
View Image Figure 12:
One-sigma errors in the time of coalescence, chirpmass and symmetric mass ratio for sources with a fixed SNR (left panels) and at a fixed distance (right panels). The errors in the time of coalescence are given in ms, while in the case of chirpmass and symmetric mass ratio they are fractional errors. These plots are for nonspinning black hole binaries; the errors reduce greatly when dynamical evolution of spins are included in the computation of the covariance matrix. Slightly modified figure from [51].
View Image Figure 13:
Distribution of measurement accuracy for a binary merger consisting of two black holes of masses m1 = 106M ⊙ and m2 = 3 × 105M ⊙, based on 10,000 samples of the system in which the sky location and orientation of the binary are chosen randomly. Dashed lines are for nonspinning systems and solid lines are for systems with spin. Figure reprinted with permission from [231]. Ⓒ External LinkThe American Physical Society.
View Image Figure 14:
The SNR integrand of a restricted (left panel) and full waveform (right panel) as seen in initial LIGO. We have shown three systems, in which the smaller body’s mass is the same, to illustrate the effect of the mass ratio. In all cases the system is at 100 Mpc and the binary’s orbit is oriented at 45° with respect to the line of sight.
View Image Figure 15:
Sky map of the gain in angular resolution for LISA observations of the final year of inspirals using full waveforms with harmonics versus restricted post-Newtonian waveforms with only the dominant harmonic, corresponding to the equal mass case (m1 = m2 = 107M ⊙, top) and a system with mass ratio of 10 (m = 107M 1 ⊙, m = 106M 2 ⊙, bottom). The sources are all at z = 1, have the same orientation (cos𝜃L = 0.2, ϕL = 3) and zero spins β = σ = 0. Figure reprinted with permission from [364]. Ⓒ External LinkThe American Physical Society.
View Image Figure 16:
By fitting the Fourier transform of an observed signal to a post-Newtonian expansion, one can measure the various post-Newtonian coefficients ψ (m ,m ),k = 0,2,3, 4,6,7 k 1 2 and coefficients of log-terms ψ5l(m1,m2 ) and ψ6l(m1, m2 ). In Einstein’s theory, all the coefficients depend only on the two masses of the component black holes. By treating them as independent parameters one affords a test of the post-Newtonian theory. Given a measured value of a coefficient, one can draw a curve in the m1m2 plane. If Einstein’s theory is correct, then the different curves must all intersect at one point within the allowed errors. These plots show what might be possible with LISA’s observation of the merger of a binary consisting of a pair of 6 10 M ⊙ black holes. In the right-hand plot all known post-Newtonian parameters are treated as independent, while in the left-hand plot only three parameters ψ0,ψ2 and one of the remainingpost-Newtonian parameter are treated as independent and the procedure is repeated for each of the remaining parameters. The large SNR in LISA for SMBH binaries makes it possible to test various post-Newtonian effects, such as the tails of gravitational waves, tails of tails, the presence of log-terms, etc., associated with these parameters. Left-hand figure adapted from [48], right-hand figure reprinted with permission from [47]. Ⓒ External LinkThe American Physical Society.