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 [306
]:
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 [303] 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
where 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) [106]
reviewed some of these phases and calculated upper limits to the GWs produced during each
phase [106
]. 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
)):
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:
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 [106] 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 [242
] opens up a new
scenario for fragmentation instabilities. Van Putten [320] 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 [239]; see
Zink et al. [348
]. 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., [138
]) 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 [143] (which assumes
) and calculated only the emission from the dominant
mode. This approach is detailed in [106
]. 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
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