Dynamical excitation of space-time modes of compact objects
aa r X i v : . [ g r- q c ] M a r Dynamical excitation of space-time modes of compact objects
Sebastiano Bernuzzi,
1, 2
Alessandro Nagar,
3, 4 and Roberto De Pietri
1, 2 Dipartimento di Fisica, Universit`a di Parma, Via G. Usberti 7/A, 43100 Parma, Italy INFN, Gruppo Collegato di Parma, Italy Institut des Hautes Etudes Scientifiques, 91440 Bures-sur-Yvette, France INFN, Sezione di Torino, Via P. Giuria 1, Torino, Italy (Dated: November 9, 2018)We discuss, in the perturbative regime, the scattering of Gaussian pulses of odd-parity gravita-tional radiation off a non-rotating relativistic star and a Schwarzschild Black Hole. We focus on theexcitation of the w -modes of the star as a function of the width b of the pulse and we contrast itwith the outcome of a Schwarzschild Black Hole of the same mass. For sufficiently narrow valuesof b , the waveforms are dominated by characteristic space-time modes. On the other hand, forsufficiently large values of b the backscattered signal is dominated by the tail of the Regge-Wheelerpotential, the quasi-normal modes are not excited and the nature of the central object cannot beestablished. We view this work as a useful contribution to the comparison between perturbativeresults and forthcoming w -mode 3D-nonlinear numerical simulation. PACS numbers: 04.30.Db, 04.40.Dg, 95.30.Sf,
I. INTRODUCTION
The pioneering works of Vishveshwara [1], Press [2] andDavis, Ruffini and Tiomno [3], unambiguously showedthat a non-spherical gravitational perturbation of aSchwarzschild Black Hole is radiated away via exponen-tially damped harmonic oscillations. These oscillationsare interpreted as space-time vibrational modes. Theproperties of these quasi-normal modes (QNMs hence-forth) of Black Holes have been thoroughly studied sincethen (see for example Refs. [4, 5, 6] and referencestherein). Relativistic stars can also have space-time vi-brational modes, the so-called w -modes [7]. These modesare purely relativistic and, contrary to fluid modes, areabsent in Newtonian theory. The fundamental w -modefrequency of a typical neutron star of radius ∼
10 kmand mass 1 . M ⊙ is expected to lie in the range of 10 ÷ ∼ − s [8].The issue of the excitation of w -modes in astrophys-ically motivated scenarios has been deeply investigatedin the literature. Andersson and Kokkotas [9] showedthat, in the odd-parity case, the scattering of a Gaussianpulse of gravitational waves off a constant density nonrotating star generates a waveform that, in close anal-ogy with the Black Hole case, is characterized by threephases: (i) a precursor, mainly related to the choice ofthe initial data and determined by the backscattering ofthe background curvature while the pulse is entering inthe gravitational field of the star; (ii) a burst; (iii) a ring-down phase dominated by w -modes, whose presence wasinferred by looking at the Fourier spectrum of the signals.Since the star is non-rotating, the signal eventually diesout with a power-law tail typical of Schwarzschild space-time [10, 11]. Allen and coworkers [12] and Ruoff [13]addressed, by means of time-domain perturbative anal-ysis, the same problem in the even-parity case, focusingon gravitational wave scattering scenarios. They consid-ered a large sample of initial configurations as well as star models of different compaction. Their main findingswere: (i) w -modes are present only for non-conformallyflat initial data (i.e., some radiative field needs to beinjected in the system) and (ii) the strength of the w -mode signal depends on the compaction of the star.These pioneering studies were later extended or refinedin Refs. [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. Inparticular, Refs. [14, 15, 16, 17] considered the scatter-ing off the star of particles moving along open orbits andrealized that the w -mode excitation strongly depends onthe orbital parameters: the closer the turning point ofthe orbit is to the star (i.e., the higher is the frequencyof the gravitational wave instantaneously emitted by theparticle), the larger is the presence of w -modes. Consis-tently, Ref. [25] showed that (modulo a simplified treat-ment of the star surface) if the source of perturbationis a spatially extended axisymmetric distribution of fluidmatter (like a quadrupolar shell) plunging on the star,the w -modes are not excited, but the energy spectrum isdominated by low-frequency contributions due to curva-ture backscattering. In addition, Ref. [26] addressed thelate-time decay of the trapped mode for ultra-compact,highly relativistic constant density stars. The presenceof trapped w -modes in stars with a first-order phasetransition (a density discontinuity) was also discussed inRef. [27].In this work we analyze the problem of w -modes ex-citation in relativistic stars (in the perturbative regime)by emphasizing the analogies with the Black Hole case.For a given odd-parity gravitational wave multipole ℓ ,we consider the scattering of Gaussian pulses of gravita-tional radiation of different width b off relativistic stars(either with constant density or with a polytropic equa-tion of state) of 1.4 M ⊙ and we contrast such signals withthose emitted by a non-rotating Black Hole of the samemass. We focus on the excitation of space-time modes asa function of the width b of the Gaussian. We find thatspace-times modes can be clearly identified only if theGaussian wave-packet is sufficiently narrow (i.e., small b ). On the other hand, for large wave-packets the internalstructure of the object is unaffected by the perturbation,tail effects are dominating and the gravitational wave-forms generated by stars or Black Holes are practicallyidentical. II. NUMERICAL FRAMEWORKA. Relativistic stars
From the spherically symmetric line element inSchwarzschild coordinates ds = − e α dt + e β dr + r (cid:0) dθ + sin θdϕ (cid:1) , (2.1)by assuming the stress energy tensor of a perfect fluid as T µν = ( p + µ ) u µ u ν + pg µν , where p is the pressure and µ the total energy density of the star, the Einstein equa-tions reduce to the Tolman-Oppenheimer-Volkoff (TOV)equations of stellar equilibrium: dmdr = 4 πr µ , (2.2) dadr = ( m + 4 πr p )( r − mr ) , (2.3) dpdr = − ( p + µ ) dαdr . (2.4)Since we are using Schwarzschild coordinates, we alsohave that e − β = 1 − m ( r ) /r , where m ( r ) is the masscontained in a sphere of radius r . This system of equa-tions needs, to be solved, the specification of an Equationof State (EoS). We use the simplest two: the constantdensity EoS and an adiabatic EoS in the form p = Kµ Γ .We consider two polytropic models (named A and B) andtwo constant energy density models (A C and B C ) withthe same compactness. All the models share the samemass M = 1 . M ⊙ , and their specific properties are listedin Table I. If not differently stated, we use geometrizedunits c = G = 1 with M ⊙ = 1. B. Odd-parity perturbations
Odd-parity linear perturbations of Black Holes andneutron stars (in the absence of external matter source)are described by a simple linear equation ∂ t Ψ (o) − ∂ r ∗ Ψ (o) + V (o) ℓ Ψ (o) = 0 , (2.5)for a master function Ψ (o) ≡ Ψ (o) ℓ that is related to themetric degrees of freedom (see for example Ref. [28]). Fora star of radius R , the potential is given by V (o) ℓ = e a (cid:18) mr + 4 π ( p − µ ) − ℓ ( ℓ + 1) r (cid:19) , (2.6) TABLE I: From up to down the rows report: the polytropicconstant K , the adiabatic index Γ, the mass of the star M ,its radius R , the central pressure p c , the central total energydensity µ c and the compaction parameter M/R , for all thestellar models considered.EoS A B A C B C K .
16 82 .
69 — —Γ 2 2 — — M .
40 1 .
40 1 .
40 1 . R .
64 9 .
10 6 .
64 9 . p c . × − . × − . × − . × − µ c . × − . × − . × − . × − M/R .
21 0 .
15 0 .
21 0 . which reduces to the standard Regge-Wheeler poten-tial for r > R , where M = m ( R ) is the total massand p = µ = 0. The latter holds also for the BlackHole case. We expressed Eq. (2.5) using a r ∗ tortoisecoordinate defined as dr ∗ /dr = exp( β − α ). This re-duces in vacuum to the Regge-Wheeler tortoise coordi-nate r ∗ = r + 2 M ln( r/ (2 M ) − E (o) ≡ X ℓ ≥ ˙ E (o) ℓ = X ℓ ≥ ( ℓ + 2)!( ℓ − | ˙Ψ (o) ℓ | , (2.7)where the over-dot stands for coordinate time derivative. C. Initial data and simulation method
For NS and Black Holes Eq. (2.5) is solved in the timedomain as an initial value problem. In the case of thepolytropic EOS, we need first to integrate numericallythe TOV equations (2.2)-(2.4) to compute the potential V (o) ℓ . For a given central pressure p c (see Table I) theTOV equations are integrated numerically (from the cen-ter outward) using a standard fourth-order Runge-Kuttaintegration scheme with adaptive step size.As initial data for (Ψ (o) , ∂ t Ψ (o) ) we set up an ingoing( ∂ t Ψ (o) = ∂ r ∗ Ψ (o) ) Gaussian pulse of tunable width b Ψ (o) = N exp (cid:2) − ( r − r ) /b (cid:3) , (2.8)where N is a normalization constant determined byequating to one the integral of Eq. (2.8) all over the ra-dial domain. This is a simple, but sufficiently general,way to represent a “distortion” of the space-time (whoseintimate origin depends on the particular astrophysicalsetting), and to introduce in the system a proper scalethrough the width of the Gaussian.Let us also summarize the basilar elements of our nu-merical procedure. Eq. (2.5) has been discretized on anevenly spaced grid (in r ∗ for the Black Hole and in r forthe star) and solved using a standard implementation ofthe second-order Lax-Wendroff method as implementedfor example in [25]. We have performed convergence testsof the code which assured a convergence factor of ∼ r ∗ = 0 .
01 and ∆ r = 0 .
015 is sufficientto be in the convergence regime. Since we have imple-mented standard Sommerfeld outgoing boundary condi-tions (see Ref. [32] for improved, non-reflecting boundaryconditions), we can’t avoid some spurious reflections tocome back from boundaries. To avoid that this effectcontaminates too much the late-time tails of the signals,we need to choose radial grids sufficiently extended, say r ∗ ∈ [ − , r ∈ [0 , III. RESULTSA. Analysis of the waveforms
We analyzed the gravitational wave response of rela-tivistic stars described by the four (two polytropic andtwo constant energy density) models in Tab. I and of aBlack Hole of the same mass to an impinging gravita-tional wave-packet of the form (2.8). We focus on thedependence of the excitation of the star w -modes (and ofthe Black Hole QNMs) ring-down on the width b . TheGaussian is centered at r = 100; the waveforms are ex-tracted at r obs = 900 ( r obs ∗ = 916) and shown versusobserver retarded time u = t − r obs ∗ . Fig. 1 exhibits thewaveforms, for Model A, Model B and the black hole for b = 2 (top), b = 8 (middle) and b = 20 (bottom). Themain panel depicts the modulus on a logarithmic scale,in order to highlight the late-time non-oscillatory tail.Let us first discuss the main features of the signal ofFig. 1, starting with the “narrow”pulse, b = 2. In thecase of the Black Hole, the ring-down has the “standard”shape dominated by the fundamental mode that is quotedin textbooks. In the case of the stars, a damped harmonicoscillations due to w -modes appears (we shall make thisstatement more precise below). The waveforms show thecommon global behavior precursor - burst - ring-down -tail . The precursor is determined by the choice of ini-tial data and by the long-range features of the potential;this implies that, until u ≃ b is increased ( b = 8), the features remain unchanged,but, although the non-oscillatory tail is not dominatingyet, the amplitude of the damped oscillation is smallerand lasting for a shorter time. A further enlargement ofthe Gaussian causes the ingoing pulse to be almost com-pletely reflected back by the “tail” of the potential, sothat the emerging waveform is unaffected by the prop-erties of the central object. The bottom panel of Fig. 1highlights this effect for b = 20: no quasi-normal oscil-lations are present. It turns out that the waveforms are
50 100 150 200 250 300 350 40010 −12 −10 −8 −6 −4 u l og | Ψ ( o ) | Model AModel BBlack Hole −100 0 100 200−2−10123 x 10 −5 u Ψ ( o ) b = 2
50 100 150 200 250 300 350 40010 −12 −10 −8 −6 −4 u l og | Ψ ( o ) | Model AModel BBlack Hole −100 0 100 200−6−4−2024 x 10 −6 u Ψ ( o ) b = 8
50 100 150 200 250 300 350 40010 −10 −8 −6 u l og | Ψ ( o ) | Model AModel BBlack HoleModel A C Model B C b = 20 FIG. 1: Dependence of the ring-down phase on the width b ofthe Gaussian pulse: for b = 2 ( top panel ) and b = 8 ( middlepanel ) the process of excitation of the space-time modes showsthe same qualitative features for the Black Hole and for thestar. The waveforms for b = 20 ( bottom panel) show there isbasically no difference between the gravitational wave signalbackscattered from a stars of Table I and from a SchwarzschildBlack Hole with the same mass. −10 ω d E / d ω b = 2b = 8b = 14b = 20
0 0.1 0.2 0.3 0.4 0.500.511.52 x 10 −11 ω d E / d ω FIG. 2: Energy spectra (from Model A) for different values of b . The maximum frequency is consistent with ω max ≃ √ /b .See text for discussion. perfectly superposed and any characteristic signature ofthe Black Hole or of the star (for any star model, seebelow) disappears. We have checked through a linearfit that the tail is (asymptotically) in perfect agreementwith the Price law: t − ℓ +3 [10, 11].The absence of QNMs for large values of b is qualita-tively explained by means of the following argument (seealso Sec. IX of Ref. [29]): in the frequency domain, theGaussian perturbation Eq. (2.8) is equivalent to a Gaus-sian of variance σ ω = √ /b and contains all frequencies.However this means that the amplitudes of the modesexcited by this kind of initial data will be exponentiallysuppressed if their frequencies are greater than the onecorresponding to three standard deviations, i.e., if theirfrequency is greater than a sort of effective maximum fre-quency given by ω b max ≃ σ ω = 3 √ /b . Generally speak-ing, we expect to trigger the space-time modes of thestar (or of the Black Hole) only when b is such that ω b max is larger than the frequency of the least damped quasi-normal mode of the system. In order to show how thisargument works, let us note that we have ω ≃ . ω ≃ . ω ≃ .
30 and ω ≃ .
21. Table II liststhe first six w -modes of Model A (for ℓ = 2): since thelowest frequency mode has ω ≃ .
29, it immediatelyfollows that for b &
14 the w -mode frequencies can’t befound in the Fourier spectrum. This argument is con-firmed by the analysis of the energy spectra, that aredepicted in Fig. 2. The frequency distribution is consis- These numbers have been computed by a frequency domaincode whose characteristics and performances are described inRefs. [20, 21, 22, 33] b E B H / E N S A (l=2)B (l=2)A C (l=2)B C (l=2)A (l=3) E B H / E N S FIG. 3: The ratio, as a function of b , of the energy releasedin gravitational waves by the Black Hole and different starmodels. tent with the value ω b max ≃ √ /b and thus the w -modescan be excited only for b . .
4. Note that the differ-ent amplitudes of the spectra in Fig. 2 are due to theconvention used for the normalization of the initial data.The same argument holds for the Black Hole. Sincewe have M = 1 . . b . . ω is smaller than the4th overtone only and, due to the correspondingly largedamping time, this is not expected to give a recognizablesignature in the waveform.On the basis of these considerations, we can summarizeour results by saying that, for our M = 1 . M ⊙ models,when b &
16, the incoming pulse is totally unaffected bythe short-range structure of the object and the signalsbackscattered by any of the stars and by the Black Holeare identical in practice. This information, deduced byinspecting the waveforms, can be synthesized by compar-ing, as a function of b and for a fixed ℓ , the energy releasedby the star ( E NS ) and by the Black Hole ( E BH ) computedfrom Eq. (2.7). Figure 3 exhibits the ratio E BH /E NS for ℓ = 2 and ℓ = 3 (the latter for Model A only). Thisquantity decreases with b because (see Ref. [1]) for small b the Black Hole, contrarily to the star, partly absorbsand partly reflects the incoming radiation. On the otherhand, the ratio tends to one for b &
16, in good numeri-cal agreement with the value of the threshold, needed toexcite the quasi-normal modes, that we estimated above.Notice that the saturation to one for ℓ = 3 occurs for val-ues of b smaller than for ℓ = 2. This is expected: in fact,the QNMs frequencies increase with ℓ and thus one needsnarrower b (and thus a larger ω b max ) to trigger space-timevibrations. TABLE II: The first four frequencies ν n and damping times τ n of w -modes (for ℓ = 2) of Model A: they have beencomputed by means of a frequency domain code describedin Ref. [20, 21, 22]. The third and fourth column of the tablelist the corresponding complex frequencies ω n − i α n in ourstandard units. We have ω n = 2 πν n M ⊙ G/c . n ν n [Hz] τ n [ µ s] ω n α n complex ℓ = 2 QNMs frequencies ω n − i α n of a M = 1 . n ν n [Hz] τ n [ µ s] ω n α n B. Identification of the w -modes We conclude this section by discussing the possibilityof identifying unambiguously the presence of w -modes inthe waveforms and in the corresponding energy spectrum.Ideally, one would like to find precise answers to the fol-lowing points: (i) understand which part of the waveformcan be written as a superposition of w -modes; (ii) howmany modes one should expect to be excited and (iii)how does this depend on b .Although these questions have been widely investi-gated in the past (see for example Chapter 4 of [5],Ref. [6] and references therein), still they have not beenexhaustively answered in the literature. The major con-ceptual problems underlying this difficulty are (i) the factthat the quasinormal-modes sets are not complete and(ii) the so called time shift problem . The former is in-trinsic in the definition of the quasinormal modes andprevents, in fact, to associate an energy to each excita-tion mode. The latter is related to the exponential decayof the quasinormal modes and it implies that, if the samesignal occurs at a later time, the magnitudes of the modeswill be larger with respect to that of the same signal oc-curred at an earlier time. As a consequence, the use ofthe magnitude of the amplitudes C n (see Eq. 3.1 below)is not a good measure of the excitation of the quasinor-mal modes. We refer to the review of Nollert [35] for athorough discussion of such problems.Beside these conceptual difficulties, from the practi-cal point of view it is however important to extract asmuch as information as possible about the quasi-normalmodes by analyzing the ringing phase of the signal. Two
120 140 160 180 200 220 240 260−5−2.502.55 x 10 −6 u Ψ ( o ) BH b=2 datan=0 Fit120 140 160 180 200 220 240 26010 −8 −5 −11 u l og | Ψ ( o ) | FIG. 4: Fits of the ring-down part of the waveform with thefundamental ( n = 0) space-time mode for a Black Hole ex-cited by a b = 2 Gaussian pulse. We show the waveformΨ (o)2 ( t ) and its absolute value on a logarithmic scale to high-light the differences with the fit. complementary methods can be used to obtain such im-portant knowledge. On the one hand, one can imple-ment the Fourier analysis, namely looking at the energyFourier spectrum in the frequency range where w -modesare expected (see e.g. Refs. [12, 15, 19]). On the otherhand, one can perform a “fit analysis”. In this case, it isassumed that, on a given interval ∆ u = [ u i , u f ], the wave-form can be written as a superposition of n exponentiallydamped sinusoids, the quasi-normal modes expansion:Ψ ℓ = X n =0 Ψ nℓ = X n =0 C n cos( ω n u + φ n ) exp( − α n u ) , (3.1)of frequency ω n and damping time 1 /α n , that are, a pri-ori, unknown [we omit henceforth the index ℓ since in thefollowing we will be focusing only on the ℓ = 2 modes].Using a non-linear fit procedure one can estimate thevalues of ( ω n , α n , C n φ n ) from the waveform. We per-form this analysis by means of a modified least-squareProny method (see e.g. the discussion of Ref. [34]) to fitthe waveforms. A feedback on the reliability of our fitprocedure is done by comparing the values of frequencyand damping time, ω nℓ and α nℓ , obtained by the fit withthose of Table II and Table III that we assume to be thecorrect ones.The typical outcome of the fit analysis, using only thefundamental mode ( n = 0), are shown in Fig. 4 for theBlack Hole with b = 2 and in Fig. 5 for the star Model Awith b = 2 (top panel) and b = 8 (bottom panel). When b = 2, for which the largest space-time mode excitationis expected, for both the star and the Black Hole the fitsshow excellent agreement with the numerical waveform atearly times, that progressively worsen due to the power-law tail contribution. The reliability of the procedure isconfirmed by the values of ω and α that we obtain fromthe fit.
130 140 150 160 170 180−5−2.502.55 x 10 −6 u Ψ ( o ) Model A b=2 datan=0 Fit130 140 150 160 170 18010 −11 −8 −5 u l og | Ψ ( o ) |
130 135 140 145 150 155−1−0.500.51 x 10 −6 u Ψ ( o ) Model A b=8 datan=0 Fit130 135 140 145 150 15510 −10 −8 −6 u l og | Ψ ( o ) | FIG. 5: Fits of the ring-down part of the waveform with thefundamental ( n = 0) space-time mode for the stellar Model Aexited by b = 2 (upper panels) and a b = 8 (bottom panels)Gaussian pulse; For each value of b we show the waveformΨ (o)2 ( t ) and its absolute value on a logarithmic scale to high-light the differences with the fit. For the Black Hole, we have ω = 0 . α =0 . .
3% and 0 .
6% fromthe “exact” values of Table III. We can thus concludethat the fundamental mode is essentially the only modeexcited for b = 2. For the star, Model A, we obtain ω = 0 . α = 0 . w -modeis generically smaller than that of an equal mass BlackHole, the ringing is shorter and it is more difficult toobtain precise quantitative statements. In this case wetried to include more modes in the template (3.1) usedfor the fit in order to precisely quantify the real contri-bution due to the presence of overtones in the signal.Unfortunately, in this case the fit procedure seems badlyconditioned and we could not obtain a sensible feedbackof the frequencies even if we clearly obtain (having moreadjustable parameters) a better fit.
120 125 130 135 140 14510 −4 −3 −2 −1 u i R BH b=2120 125 130 135 140 14510 −1 −2 −3 −4 u i R Model A b=2
FIG. 6: The residual
R ≡ − Θ (See Eq. 3.2) of the fits of thewaveform of the response to a b = 2 Gaussian pulse of a BlackHole (upper panel) and stellar Model A (bottom panel) as afunction of the initial time ( u i ) of the fitting window aroundits best values that it is u i = 129 for a Black Hole and u i = 131for Model A. We have found that the choice of the time window toperform this analysis has a strong influence on the re-sult of the fit. This choice is delicate and it is relatedto the aforementioned problem of the time shift. Ideally,the window should start with the ring-down (i.e., at theend of the burst ) and it must be both sufficiently narrow,in order not to be influenced by the non-oscillatory tail,and sufficiently extended to include all the relevant in-formation. There are no theoretical ways to predict orestimate the correct window, but some systematic proce-dures have actually been explored in the literature. Wedecided to use a method very similar to the one discussedin details in Ref. [30]: it consists in setting u f at the endof the oscillatory phase, which is clearly identifiable ina logarithmic plot, and choosing the initial time of thewindow u i such as to minimize the difference between thereal data (Ψ data j ) and the waveform synthesized from theresults of the fitting procedure (Ψ fit j ). This difference is
119 124 129 134 1380.2620.2630.2640.2650.2660.2670.2680.269 ω u i BH b=2 122 127 132 13700.10.20.30.40.50.6 ω u i Model A b=2119 124 129 134 1380.0550.060.0650.070.0750.080.0850.09 α u i BH b=2 122 127 132 1370.10.20.30.40.50.60.70.8 α u i Model A b=2
FIG. 7: Determination of the best window for the fit of the Black Hole waveform and Model A ( b = 2). The initial time u i ischosen so to minimize the residual R between the data and the fit. For the Black Hole we obtain u i = 129 ( u f = 254), whilefor Model A we have u i = 131 ( u f = 174). The horizontal line indicates the “exact” values of the considered model reported inTable II and Table III. estimated by means of the following “scalar product”Θ(Ψ data , Ψ fit ) ≡ P j Ψ data j Ψ fit j qP j (Ψ data j ) qP j (Ψ fit j ) (3.2)whose result Θ is a value in the interval [0 ,
1] that it isexactly one when the two time series are identical (perfectfit). Fig. 6 shows such a determination for the Black Hole(top panel) and Model A (bottom panel) with b = 2:both curves exhibit a clear minimum of the quantity R ≡ − Θ at, respectively, u i = 129 and u i = 131. The timewindow extends to u f = 254 (Black Hole) and u f = 174(Model A), respectively.As can be seen in Fig. 7 one has that even a smallchange of initial time u i of the window used produces sen-sible variation of the estimated values of the frequencyand the damping time of the fundamental mode. How-ever, it should be noticed that the estimated values ob-tained for the best window are those in best agreementwith the expected values. We finally repeat the analysis for the datasets relativeto wider Gaussian pulses. Focusing on the representa-tive b = 8 case, we find essentially the same picture withtwo main differences. First, the “global” quality of thefit, given by R is less good than in the b = 2 case; thisis particularly evident for the star, where the fitted fre-quencies differ from the “exact” ones by more than 10%.Second, the fitting window becomes narrower and nar-rower as b is increased (see column seven of Table IV),thereby the the fit analysis quickly becomes meaningless.For the reasons outlined above we have found this pro-cedure not as effective as we hoped. We think that theproblem of the unambiguous determination of the “right”time interval for the fit and of the presence and quantifi-cation the overtones in numerical data deserves furtherconsiderations. TABLE IV: The results of the fit of the fundamental mode forthe Black Hole and for the stellar Model A in the responseto Gaussian pulse with b = 2 and b = 8 for the best fitwindows [ u i , u f ] determined using the minimum of the residual R ≡ − Θ criteria (see Fig. 6). The reported error referto relative difference between the fitted values and referencevalues reported in Table II and Table III.Model b ω δω % α δα % [ u i , u f ] R A 2 0.2739 7 0.1636 8 [131,174] 2 × − A 8 0.3396 15 0.1302 13 [131,154] 1 × − BH 2 0.2660 0.3 0.0631 0.6 [129,254] 4 × − BH 8 0.2614 0.2 0.0591 0.7 [129,229] 5 × q − IV. CONCLUSIONS
In this paper we have studied numerically the scatter-ing of odd-parity Gaussian pulses of gravitational radia-tion off relativistic stars and Black Holes. We have foundthat the excitation of w -modes and black hole QNMs oc-curs basically in the same way for both objects: pulses ofsmall b (high frequencies) can trigger the w -modes, whilefor large b (low frequencies) one can only find curvaturebackscattering effects and non-oscillatory tails. When w -modes are present, we have shown that both frequency-domain (energy spectrum) and time-domain (fit to a su-perposition of w -modes) analysis are useful to understandthe mode content of the waveforms; however, our studyalso indicates that it is difficult to single out precisely thecontribution of each mode, since the fundamental modealways dominates the signal and a clear identification ofthe overtones is lacking in the case of excitation inducedby the scattering of odd-parity Gaussian pulses of gravi-tational radiation. The inspiring idea of this paper was to understand theorigin of the dynamical excitation of w -modes in a sim-ple, but rather general, setting, where it is possible todo many, quick and controllable high-accuracy numeri-cal simulations with a tunable “source”. Our expecta-tion is that the main features of the process of w -modeexcitation (i.e., its dependence on the intrinsic frequencycontent of the initial data) that we have highlighted inthe perturbative regime are sufficiently robust to survive qualitatively also in the full 3D-nonlinear case that needsthe solution of the full set of Einstein equations. An en-couraging motivation for hoping so is that in other phys-ical settings, like for example the merger of two equalmass Black Holes, the perturbative analysis of the early70s [3] in the extreme mass ratio limit was able to singleout the important physical elements (i.e., the presence ofQNMs), sketching a picture that has been later refined,but substantially confirmed, by numerical relativity sim-ulations [36, 37, 38, 39, 40]. Acknowledgments
Computation performed on the
Albert
Beowulfclusters at the University of Parma. We grate-fully thank V. Ferrari and L. Gualtieri for com-puting for us the w -mode frequencies of Table II.The commercial software Matlab TM has been used formost of the analysis reported here. Software im-plementing the Prony methods is freely available at . SB hasbeen supported by a Marie-Curie fellowship during the General Relativity Trimester on Gravitational Waves,Astrophysics and Cosmology at Institut Henri Poincar´e(Paris), where part of this work was done. The activityof AN at IHES is supported by an INFN fellowship. ANalso gratefully acknowledges support of ILIAS. [1] C.V. Vishveshwara, Nature, , 936 (1970).[2] W.H. Press, Astrophys. J. Letts. , L105 (1971).[3] M. Davis, R. Ruffini and J. Tiomno, Phys. Rev. D ,2932 (1972).[4] S. Chandrasekhar The Mathematical Theory of BlackHoles ,Oxford University Press Inc., New York, 1983.[5] V.P. Frolov and I.D. Novikov,
Black Hole Physics ,Kluwer Academic Publishers, 1998.[6] K. Kokkotas, and B.G. Schmidt, Living Reviews 1999-2(1999).[7] K. Kokkotas and B.F. Schutz, Mon. Not. R. Astr. Soc. , 119 (1992).[8] N. Andersson and K. D. Kokkotas, Lect. Notes Phys. , 255 (2004)[9] N. Andersson and K.D. Kokkotas, Phys. Rev. Lett. ,4134 (1996).[10] R.H. Price, Phys. Rev. D , 2419 (1972).[11] R.H. Price, Phys. Rev. D , 2439 (1972).[12] G. Allen, N. Andersson, K.D. Kokkotas and B.F. Schutz,Phys. Rev. D , 124012 (1998). [13] J. Ruoff, Phys. Rev. D , 064018 (2001).[14] K. Tominaga, M. Saijo and K. I. Maeda, Phys. Rev. D , 024004 (1999).[15] V. Ferrari and K.D. Kokkotas, Phys. Rev. D , 107504(2000).[16] K. Tominaga, M. Saijo and K. I. Maeda, Phys. Rev. D , 124012 (2001)[17] J. Ruoff, P. Laguna and J. Pullin, Phys. Rev. D ,064019 (2001).[18] A. Passamonti, M. Bruni, L. Gualtieri and C.F. Sopuerta,Phys. Rev. D , 024022 (2005).[19] A. Passamonti, M. Bruni, L. Gualtieri, A. Nagar andC.F. Sopuerta, Phys. Rev. D , 084010 (2006).[20] J. A. Pons, E. Berti, L. Gualtieri, G. Miniutti and V. Fer-rari, Phys. Rev. D , 104021 (2002);[21] L. Gualtieri, E. Berti, J. A. Pons, G. Miniutti and V. Fer-rari, Phys. Rev. D , 104007 (2001);[22] O. Benhar, E. Berti and V. Ferrari, Mon. Not. Roy. As-tron. Soc. , 797 (1999).[23] N. Andersson, Phys. Rev. D , 1808 (1995). [24] N. Andersson, Phys. Rev. D , 468 (1997).[25] A. Nagar, G. Diaz, J.A. Pons and J.A. Font, Phys. Rev.D , 124028 (2004)[26] V. Pavlidou, K.T. Tassis, T.W. Baumgarte, andS.L. Shapiro, Phys. Rev. D , 084020 (2000).[27] Z. Andrade, Phys. Rev. D , 124002 (2001)[28] A. Nagar and L. Rezzolla, Class. Quant. Grav. , R167(2005); erratum, ibid. , 4297 (2006).[29] N. Andersson, Phys. Rev. D , 353 (1995).[30] E. N. Dorband, E. Berti, P. Diener, E. Schnetter andM. Tiglio, Phys. Rev. D , 102503 (2005)[arXiv:gr-qc/0507140]. [33] V. Ferrari and L. Gualtieri, private communication.[34] E. Berti,V. Cardoso, Jose A. Gonzalez and U. Sperhake,Phys. Rev. D 75, 124017 (2007)[35] Hans-Peter Nollert, Class. Quantum Grav. 16 R159,(1999)[36] F. Pretorius, Phys. Rev. Lett. , 121101 (2005)[37] M. Campanelli, C. O. Lousto, P. Marronetti andY. Zlochower, Phys. Rev. Lett. , 111101 (2006)[arXiv:gr-qc/0511048].[38] J. G. Baker, J. Centrella, D. I. Choi, M. Koppitzand J. van Meter, Phys. Rev. Lett. , 111102 (2006)[arXiv:gr-qc/0511103].[39] J. G. Baker, M. Campanelli, F. Pretorius and Y. Zlo-chower, Class. Quant. Grav.24