2 Gravitational Wave Emission

Most studies of GW emission use a multipole expansion of the perturbation h μν to a background spacetime b gμν. The lowest (quadrupole) order piece of this field is [306Jump To The Next Citation Point]:

[ ]TT TT 2G- d2- -8-G- -d2 hjk = dc4 dt2ℐjk(t − r) + 3d c5𝜖pq(jdt2𝒮k )p(t − r)nq . (1 )
Here, ℐjk and 𝒮jk are the mass and current quadrupole moments of the source, d is the distance from the source to the point of measurement, 𝜖ijk is the antisymmetric tensor, and nq is the unit vector pointing in the propagation direction. Parentheses in the subscripts indicate symmetrization over the enclosed indices, and the superscript TT indicates that one is to take the transverse-traceless projection; G is Newton’s gravitational constant, and c is the speed of light.

Most GW estimates are based on Equation (1View Equation). 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 ℐjk as a moment of the spacetime rather than as a mass moment [310, 162Jump To The Next Citation Point]). 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 (1View Equation) 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

∫ [ ] ℐjk = d3x ρ xjxk − 1r2δjk , (2 ) 3
where ρ is the mass density, and δ = 1 jk for j = k and 0 otherwise. The second term in the integrand ensures that the resulting tensor is trace free.

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) [106Jump To The Next Citation Point] reviewed some of these phases and calculated upper limits to the GWs produced during each phase [106Jump To The Next Citation Point]. Recent studies have confirmed these upper limits, typically predicting results between 5 – 50 times lower than the secure FHH upper limits [165Jump To The Next Citation Point, 228Jump To The Next Citation Point]. 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 (15View Equation) – (19View Equation)):

∘ ---- < (rh+ )2 >1 ∕2 ≈ G ∕c4 8∕3 (apxp + v2p)mp, (3 )
where ap is the acceleration, xp is the position (providing a lever arm for the GW emission), vp is the velocity, and m p is the mass of the parcel of matter driving the GW emission. If we limit our estimate to the acceleration term, we find
< (rh+ )2 >1∕2 ≈ 50(ap∕6 × 1011 cm s−2)(xp∕30 km )(mp ∕0.1M ⊙). (4 )
For a 0.1M ⊙ asymmetry, where the matter at 30 km is accelerated from − 0.1c to 0.1c in 10 ms (typical for a core-collapse bounce), the signal would be roughly 16 × 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 0.01c. As such, we expect convective motion alone to be an order of magnitude lower than bounce:

2 1∕2 9 −2 < (rh+ ) > ≈ 5(ap∕6 × 10 cm s )(xp∕100km )(mp ∕0.3M ⊙ ). (5 )
That is, for typical convective regions, we expect a signal of less than 1.6 × 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 [106Jump To The Next Citation Point] 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 [245Jump To The Next Citation Point, 113Jump To The Next Citation Point, 282Jump To The Next Citation Point]. However, the argument that massive accretion disks act as gamma-ray burst (GRB) engines [242Jump To The Next Citation Point] 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 Ωc ∕πG Σ < 1 s [316, 123] is a necessary, but not sufficient condition [239]; see Zink et al. [348Jump To The Next Citation Point]. 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):

P = 2 ∕5G4∕c5m5 ∕r5, (6 ) fragment
∘ ---- hfragment = 8∕5G2 ∕c4m2∕ (rd ), (7 )
where G is the gravitational constant, c is the speed of light, m is the mass of the fragments, r is their separation, and d is the distance to the detector. In the black-hole forming case, when the “horizons” of the two fragments touch, the power radiated reaches a maximum: 5 57 −1 P = c ∕(80G ) ∼ 10 ergs s. This is a strong source, if simulations confirm that such fragments exist. For black holes, fragmentation will lead to clump accretion and black hole ringing.

Considerably more effort has been focused on bar modes, or more generally, non-axisymmetric instabilities. Classical, high β (where β is the ratio of the rotational to potential energy ≡ T ∕|W |), bar modes can be separated into two classes: dynamical instabilities and secular instabilities. The classical analysis predicts that β ≳ 0.27 is needed for dynamical instabilities and (β ≳ 0.14) is required for secular instabilities. Collapse calculations of many modern progenitors (e.g., [138Jump To The Next Citation Point]) followed through collapse calculations [245Jump To The Next Citation Point, 113Jump To The Next Citation Point, 69Jump To The Next Citation Point] 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 [338Jump To The Next Citation Point, 340Jump To The Next Citation Point], binary scenarios [114Jump To The Next Citation Point, 240Jump To The Next Citation Point, 104Jump To The Next Citation Point, 319Jump To The Next Citation Point, 42Jump To The Next Citation Point], and the collapse of merging white dwarfs [341Jump To The Next Citation Point, 339Jump To The Next Citation Point, 342Jump To The Next Citation Point, 118Jump To The Next Citation Point, 63Jump To The Next Citation Point, 61Jump To The Next Citation Point]. Second, calculations including differentially-rotating cores also discovered that, with differential rotation, instabilities can occur at much lower values for β, as low as 0.01 [278Jump To The Next Citation Point, 279Jump To The Next Citation Point]. 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 2r:

32G Pbar = ----m2r4 ω6 (8 ) 45c5
∘ ------ hbar = 32∕45G ∕c4mr2 ω2∕d. (9 )

Lastly, there has been considerable discussion about the existence of r-modes in the cooling proto neutron star [4Jump To The Next Citation Point, 96Jump To The Next Citation Point] . FHH estimated an upper limit for the r-mode signature by using the method of Ho and Lai [143] (which assumes αmax = 1) and calculated only the emission from the dominant m = 2 mode. This approach is detailed in [106Jump To The Next Citation Point]. If the neutron star mass and initial radius are taken to be 1.4M ⊙ and 12.53 km, respectively, the resulting formula for the average GW strain is

( ) −24 ( ν s ) 20 Mpc h (t) = 1.8 × 10 α ------ -------- , (10 ) 1 kHz d
where α is the mode amplitude and νs is the spin frequency. Recent studies suggest that viscosities may be quite high in a cooling proto neutron star, making this source untenable [159Jump To The Next Citation Point, 158Jump To The Next Citation Point, 248Jump To The Next Citation Point, 249Jump To The Next Citation Point, 181Jump To The Next Citation Point, 178Jump To The Next Citation Point, 132Jump To The Next Citation Point, 5Jump To The Next Citation Point]. We will review this argument, and some of the recent progress in estimating the viscosity in a neutron star in Section 4.5.

 2.1 Making numerical estimates

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