Most studies of GW emission use a multipole expansion of the perturbation to a background spacetime . The lowest (quadrupole) order piece of this field is :
Most GW estimates are based on Equation (1). When bulk mass motions dominate the dynamics, the first term describes the radiation. For example, this term gives the well-known “chirp” associated with binary inspiral. It can be used to model bar-mode and fragmentation instabilities. At least conceptually, this term also applies to black hole ringing, provided one interprets as a moment of the spacetime rather than as a mass moment [310, 162]). In practice, ringing waves are computed by finding solutions to the wave equation for gravitational radiation  with appropriate boundary conditions (radiation purely ingoing at the hole’s event horizon, purely outgoing at infinity). The second term in Equation (1) gives radiation from mass currents, and is used to calculate GW emission due to the r-mode instability.
When the background spacetime is flat (or nearly so) the mass and current moments have particularly simple forms. For example, in Cartesian coordinates the mass quadrupole is given by
There are several phases during the collapse (and resultant explosion) of a star that produce rapidly changing quadrupole moments: both in the baryonic matter and in the neutrino emission. We will break these phases into four separate aspects of the core-collapse explosion: the bounce of the core, the convective phase above the proto neutron star, cooling in the neutron star, and formation phase of a black hole. Scientists have focused on different phases at different times, arguing that different phases dominated the GW emission. These differences are primarily due to different initial conditions. A specific stellar collapse will only pass through a subset of these phases and the magnitude of the GW emission from each phase will also vary with each stellar collapse. Fryer, Holz & Hughes (hereafter FHH)  reviewed some of these phases and calculated upper limits to the GWs produced during each phase . Recent studies have confirmed these upper limits, typically predicting results between 5 – 50 times lower than the secure FHH upper limits [165, 228]. Before we discuss simulations of GWs, we review these FHH estimates and construct a few estimates for phases missed by FHH.
Those collapsing systems that form neutron stars (all but the most massive stellar collapses) will emit a burst of GWs at bounce if there is an asymmetry in the bounce (either caused by rotation or by an asymmetry in the stellar core prior to collapse). If we represent the matter asymmetry in the collapse by a point mass being accelerated as the core bounces, we can estimate the GW signal from the bounce phase (using, for example, Equations (15) – (19)):× 10–22 at 10 kpc. Although one could imagine increasing this signal by including the velocity term, it is unlikely that we will be able to increase this value by more than an order of magnitude.
This formulation can also be used to estimate the GW emission for convection. After bounce, convection begins to grow within and on top of the proto neutron star2. In this extreme case, the GW signal convection can once again be estimated by a point source accelerating across the convective regime. Here the acceleration of the matter occurs over a 100 ms timescale and the velocities are peaking at roughly . As such, we expect convective motion alone to be an order of magnitude lower than bounce:× 10–22 at 10 kpc. Any post-bounce signal stronger than this result is likely to arise from additional instabilities such as bar modes.
FHH focused on three emission sources after the bounce. In the convective regime, they studied the development of two different instabilities: fragmentation and bar modes. Bar modes can also develop in the neutron star as it cools. FHH  also estimated the r-mode strength in neutron stars. Let’s review these estimates.
Fragmentation can only occur in very rapidly-rotating models. Currently, the predicted rotation rates of stellar models are simply not fast enough to develop such instabilities [245, 113, 282]. However, the argument that massive accretion disks act as gamma-ray burst (GRB) engines  opens up a new scenario for fragmentation instabilities. Van Putten  has argued that fragmentation-type instability can occur in black-hole–forming collapses, if it is rotating sufficiently fast to form a massive disk. Thus far, simulations of such disks have developed Rossby-like instabilities, but these instabilities do not lead to true fragmentation. Requiring [316, 123] is a necessary, but not sufficient condition ; see Zink et al. . Assuming they can occur, GW estimates of fragmentation provide a strong upper limit to the GW signal. In Keplerian orbits, with equal massed fragments, this limit is (FHH):
Considerably more effort has been focused on bar modes, or more generally, non-axisymmetric instabilities. Classical, high (where is the ratio of the rotational to potential energy ), bar modes can be separated into two classes: dynamical instabilities and secular instabilities. The classical analysis predicts that is needed for dynamical instabilities and () is required for secular instabilities. Collapse calculations of many modern progenitors (e.g., ) followed through collapse calculations [245, 113, 69] suggest that such “classical” instabilities do not occur in nature.
However, advances in both progenitors and non-axisymmetric instabilities have revived the notion of bar modes. First, the search for progenitors of GRBs has led stellar and binary modelers to devise a number of collapse scenarios with rapidly-rotating cores: mixing in single stars [338, 340], binary scenarios [114, 240, 104, 319, 42], and the collapse of merging white dwarfs [341, 339, 342, 118, 63, 61]. Second, calculations including differentially-rotating cores also discovered that, with differential rotation, instabilities can occur at much lower values for , as low as 0.01 [278, 279]. We will discuss these new discoveries in detail in Section 4.3. FHH estimated the upper limits of the GW signals from bar-instabilities assuming a simple single bar of length :
Lastly, there has been considerable discussion about the existence of r-modes in the cooling proto neutron star [4, 96] . FHH estimated an upper limit for the r-mode signature by using the method of Ho and Lai  (which assumes ) and calculated only the emission from the dominant mode. This approach is detailed in . If the neutron star mass and initial radius are taken to be and 12.53 km, respectively, the resulting formula for the average GW strain is[159, 158, 248, 249, 181, 178, 132, 5]. We will review this argument, and some of the recent progress in estimating the viscosity in a neutron star in Section 4.5.
Living Rev. Relativity 14, (2011), 1
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