3.3 The PerBaCCo
3.3.1 Initial data
In principle the choice of the initial data for the perturbation equations should take into account, at least approximately, the astrophysical scenario in which the neutron star is born. Such scenario could be, for example the gravitational collapse or the merger of two neutron stars. Only long-term simulations in full general relativity can investigate highly nonlinear and non-isotropic system until they settle down in a, almost spherical, quasi-equilibrium configuration.
A perturbative analysis could then start from this point once the metric and matter tensors had been projected along the corresponding (tensorial) spherical harmonics. This approach, in principle possible, is however beyond the scopes of the present work.
Inspired by previous perturbative calculations [6,169], we implement in the PerBaCCo different kinds of initial data such that they are the simplest, well-posed and involve perturbations of both the fluid and/or the metric quantities.
In the case of the even parity perturbations 3 different initial excitations can be evolved:
1. Conformally Flat Initial Data. We set S(0, r) = 0 and give a fluid pertur-bation of type:
H(0, r) = Ar R
`−1 sin
π(n + 1)r R
. (3.3.1)
The function k(0, r) is computed consistently solving the Hamiltonian con-straint. The profile of H in Eq. (3.3.1) is chosen in order to approximate the behavior of an enthalpy eigenfunction with n nodes. In this way, only some modes can be (prominently) excited. To focus for example the prin-cipal fluid modes one could chose a zero-nodes initial data setting n = 0.
2. Radiative Initial Data. We set k(0, r) = 0 and H(0, r) as Eq. (3.3.1).
The function S(0, r) is computed consistently solving the Hamiltonian constraint.
3. Scattering-like Initial Data. We set H(0, r) = 0 and Ψ(e)(0, r) as a Gaus-sian pulse:
Ψ(e)(0, r) = A exp
−(r − rc)2 b2
, (3.3.2)
The functions k(0, r) and S(0, r) are computed consistently from Eq. (3.2.10) and Eq. (3.2.11).
Initial data of type 1 and 2 are chosen to be time-symmetric (S,t = k,t = H,t = 0). On the one hand, this choice can be physically questionable because the system has an unspecified amount of incoming radiation in the past. On the other hand, it is the simplest choice and guarantees that the momentum
Chapter 3. Perturbations of non-rotating stars 31
30 35 40 45 50 55
0 1 2 3 4 5
x 10−4
r Ψ(e)(0,r)
Exact
4th−order, 100 pts 2nd−order 500 pts 2nd−order 100 pts
Figure 3.1: Initial data of type 3 (See text) for a perturbative evolution. Convergence of the Zerilli-Moncrief function as a function of the radial coordinate. Only high resolution and/or an high accuracy in the computation in the (finite differencing) derivative ∂rk give reliable initial data.
constraints are trivially satisfied and only the Hamiltonian constraint needs to be used for the setup. The Gaussian in type 3 initial data is in-going, i.e.
Ψ,t= Ψ,r∗, and the derivatives of the other variables are computed consistently with this choice.
In the case of odd-parity perturbations we start the evolution using an in-going narrow Gaussian as in Eq. (3.3.2) for Ψ(o)(0, r). We stress that this is a simple, but sufficiently general, way to represent a “distortion” of the space-time (whose intimate origin depends on the particular astrophysical setting), and to introduce in the system a proper scale through the width of the Gaussian.
In some situations, the convergence of the Zerilli-Moncrief computed from Eq. (3.2.9) can be particularly delicate. For example, in case of type 3 initial data, a very accurate computation of the radial derivative k,ris needed to obtain an accurate and reliable Ψ(e) from k and S. Fig. 3.1 summarizes the kind of problem that one can find computing the Zerilli-Moncrief function too naively.
It refers to Ψ(e)(r) for (`, m) = (2, 0) at t = 0. We consider (for simplicity of discussion) a polytropic model, i.e. p = KρΓ with K = 100, Γ = 2 and ρc= 1.28×10−3with type 3 initial data (A = 0.1, b = M and rc= 40). We fixed Ψ(e) by Eq. (3.3.2), we computed χ and k from Eq. (3.2.10) and Eq. (3.2.11),
∂rk numerically using finite differencing schemes and then we reconstructed
32 3.3. The PerBaCCo
Ψ(e) via Eq. (3.2.9). Fig.3.1 shows that, for low resolution (Ji = 100), using a second-order finite-differencing standard stencil to compute ∂rk is clearly not enough, as the reconstructed (thicker dashed-dot line) and the “exact” (solid line) Ψ(e)are very different. Increasing the resolution (to 500 points) improves the agreement, which is however not perfect yet (thinner dashed-dot line). A visible improvement is obtained using higher order finite-differencing operators:
the figure shows that a 4th-order operator (already in the low-resolution case with Ji = 100 points) is sufficient to have an accurate reconstruction of the Zerilli-Moncrief function.
The conclusion is that one needs to use at least 4th-order finite-differencing operators to compute accurately Ψ(e)from χ and k. If this is not done, the re-sulting function is not reliable and it can’t be considered a solution of Eq. (3.2.6).
Typically, we have seen that, when this kind of inaccuracy is present, the am-plitude of (part of) the Zerilli-Moncrief function (usually the one related to the w-mode burst) grows linearly with r instead of tending to a constant value for r → ∞. A very detailed discussion of this problem will be presented in Sec.6.6.
Chapter 4
Gravitational waves from neutron star linear
oscillations
The pioneering works of Vishveshwara [205], Press [160] and Davis, Ruffini and Tiomno [75], unambiguously showed that a non-spherical gravitational pertur-bation of a Schwarzschild black hole is radiated away via exponentially damped harmonic oscillations. These oscillations are interpreted as space-time vibra-tional modes. The properties of these quasi-normal modes (QNMs henceforth, see App. A) of black holes have been thoroughly studied since then, see for example Refs. [68,93,115] and references therein.
Gravitational waves from relativistic stars are also characterized by the sign of the proper mode of oscillations the emitting star. The QNMs frequencies carry information about the internal composition of the star and, once detected, they could, in principle, be used to put constraints on the values of mass and radius and thus on the Equation of State (EOS) of a NS [14]. In general these non-spherical oscillations are of two types [187,115,147]: fluid modes, related to the fluid pulsations and which have a Newtonian counterpart, and spacetime or curvature modes, which exist only in relativistic stars and are weakly coupled to matter. The former in particular are classified [114], as the fundamental or f -mode with typical frequencies around νf ∼ 3 kHz and damping time τf ∼ 0.1 s, pressure or p-modes whose restoring force is pressure ( νp1 > 4 kHz, τp1 ∼ 0.6 s) and gravity or g-modes whose restoring force is gravity (present only in non-isentropic star, T 6= 0) and rotational or r-modes typical of rotating stars.
The latter are spacetime vibrational modes, the so-called w-modes [116]. The fundamental w-mode frequency of a typical NS is expected to lie in the range of 10 ÷ 12 kHz and to have a damping time of ∼ 10−4 s [15].
In this Chapter we present results obtained by PerBaCCo simulations: var-ious kind of initial data for odd and even-parity perturbations have been evolved for 47 NS models computed from 10 different EOS. The GW computed have been studied in great details and in different aspect: comparing star w-modes with black hole QNMs, investigating the effect of the EOS and the compact-ness on proper frequencies, and exploring techniques to extract them from the waves. These results extend and complete the previous work by Allen et al. [6]
34 4.1. Analysis methods
and Ruoff [169]. Moreover, the results obtained prove the performances of the PerBaCCo and pose the basis to be used as test-bed for nonlinear code.
4.1 Analysis methods
Once obtained a waveform Ψ(t) from a numerical simulation it is in general necessary to understand the physical (dynamical) origin of the different parts of the wave, and in particular to “quantify” the signal, i.e. to extract amplitude, phase and frequencies.
Two complementary methods can be used to obtain such important knowl-edge. On the one hand, one can use the Fourier analysis and look at the energy spectrum of the wave (See Eq. (3.2.18)) On the other hand, one can perform a “fit analysis” based on specific template for the wave. In the case of stellar oscillations around the equilibrium configuration, the wave are expected to be characterized by one or more frequencies related to the proper modes of the star, i.e. the QNMs. As a consequence it is natural to assume that, on a given interval
∆t = [ti, tf], the waveform can be written as a superposition of N exponentially damped sinusoids, the QNMs expansion:
Ψ(t) =
N
X
n=0
Ancos (ωnt + φn) exp (−αnt) , (4.1.1)
of frequencies νn = ωn/2π and damping times τn ≡ 1/αn. Using a non-linear fit procedure one can estimate the values of ωn, αn, An and φn from the signal Ψ(t) that are, a priori, unknown. Typically the method used is the standard least-squares fit based on the reduced χ2 minimization [161]. Alternatively the fit can be performed with different techniques like, for example, the least-square Prony methods, already used to this specific purpose (see e.g. Ref. [45]).
The outcome of a numerical simulation is a time series fj, i.e. a discrete set of (floating-point) numbers describing the the physical quantity Ψ(t). After a Fourier or fit analysis, i.e. after knowing the proper frequencies of the system, the questions that emerge in the analysis of the waveforms are: “How far” are two signals? What is the “global difference” between two signals? (and also
“How good” is the fit?). The answer requires a way to quantify the “distance”
between two time series. To this aim, and taking into account that we are typically dealing with functions of the type in Eq. (4.1.1), we consider the (l2) scalar product
Θ(f, g) ≡
P
jfjgj qP
j(fj)2 qP
j(gj)2
, (4.1.2a)
from which one can define the “residual”
R(f, g) ≡ 1 − Θ(f, g) (4.1.2b)
which is bounded in the interval [0, 1], and it is exactly 0 when the two time series are identical. This residual gives a measure of a “relative” distance be-tween fj and gj. As a second measure of the global distance (agreement) of two time series we use the l∞ distance:
D(f, g) ≡ max
j |fj− gj| , (4.1.3)
Chapter 4. GW from NS oscillations: linear simulations 35
that gives the maximum, or “absolute”, difference between fj and gj.
In the following Sections (and Chapters) we use systematically and all to-gether these methods to investigate the waveforms explicitly computed from time-domain code, showing how, in most of all the cases, they are simple and powerful instruments to extract reliable informations from gravitational wave signals.